Big Ideas 5th Grade Chapter 1 Place Value Concepts Math Book Answer Key

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Lesson 1: Place Value Patterns

• Lesson 1.1 Place Value Patterns
• Place Value Patterns Homework & Practice 1.1

Lesson 2 Place Value with Whole Numbers

• Lesson 1.2 Place Value with Whole Numbers
• Place Value with Whole Numbers Homework & Practice 1.2

Lesson 3 Patterns and Powers of 10

• Lesson 1.3 Patterns and Powers of 10
• Patterns and Powers of 10 Homework & Practice 1.3

Lesson 4 Decimals to Thousandths

• Lesson 1.4 Decimals to Thousandths
• Decimals to Thousandths Homework & Practice 1.4

Lesson 5 Place Value with Decimals

• Lesson 1.5 Place Value with Decimals
• Place Value with Decimals Homework & Practice 1.5

Lesson 6 Compare Decimals

• Lesson 1.6 Compare Decimals
• Compare Decimals Homework & Practice 1.6

Lesson 7 Round Decimals

• Lesson 1.7 Round Decimals
• Round Decimals Homework & Practice 1.7

Performance Task

• Place Value Concepts Performance Task
• Place Value Concepts Activity
• Place Value Concepts Chapter Practice

Lesson 1.1 Place Value Patterns

Explore and Grow

Write the whole number represented by each base ten block. Then use the base ten blocks to complete the table.

Answer:

Rod is 10 times as much as Unit

The flat is 10 times as much as Rod

Cube is 3 times as much as Flat

Unit is 1/10 of Rod

Rod is 1/10 of Flat

The flat is 3/10 of Cube

Reasoning
Describe the patterns you see in a number as you move from one place value position to another place value position.

Answer: As we move from one place value position to another value position, The place value of a digit increase by ten times as move from the left.

Think and Grow: Place Value Patterns

You can use a place value chart to help write numbers that are 10 times as great as a number or \(\frac{1}{10}\) of a number.

Show and Grow

Question 1.
Complete the statements.

Apply and Grow: Practice

Use a place value chart to answer the question.
Question 2.
What number is 10 times as great as 6,000?
Answer:60,000

If 6,000 is multiplied by 10 times it becomes 60,000

Question 3.
What number is \(\frac{1}{10}\) of 300?
Answer:30

300 x 1/ 10 is 30

Question 4.
80 is 10 times as great as what number?
Answer: 8

if 8 is multiplied by 10 times it becomes 80

Question 5.
40,000 is \(\frac{1}{10}\) of what number?
Answer:4,00,000

4,00,000 x 1/10 is 40,000

The number is 4,00,000

Complete the table.
Question 6.

Answer:

1. 100 ,1 ,

10,  10 times as number is 10 x 10 = 100

1/10 of 10 is 1

2. 4000,40

400, 10 times as number is 400 x 10 = 4000

1/10 of 400 is 40

3. 70,000,700

7,000, 10 times as number is 7,000 x 10 =70,000

1/10 of 7,000 is 700

4. 500,000,    5,000

50,000,10 times as number is 50,000 x 10 = 5,00,000

1/10 of 50,000 is 5,000

Question 7.

Answer:

1. 2,000, 20

200,  10 times as number is 200 x 10 = 2,000

1/10 of 200 is 20

2. 3,00,000,   3,000

30,000,  10 times as number is 30,000 x 10 = 3,00,000

1/10 of  30,000 is 3,000

3. 900, 9

90, 10 times as number is 90 x 10 = 900

1/10 of 90 is 9

4. 80,000, 800

8,000,  10 times as number is 8,0,0 x 10 = 80,000

1/10 of 8,000 is 800

Question 8.
Patterns
Describe the relationship between any place value position and the next greater place value position.
Answer:

The relation between any place value position and next greater place value position increases ten times as we move.

Number Sense
Write whether the statement is true or false. If false, explain why.
Question 9.
600 is 100 times as great as 60,000.
______
Answer: True

600 X 100 = 60,000

Question 10.
9,000 is 1,000 times as great as 9.

_____
Answer: True

9 X 1,000 = 9,000

Think and Grow: Modeling Real Life

Example
Which state is about 10 times larger than Georgia?
The approximate land area of Georgia is 60,000 square miles.
Use a place value chart to find the number that is 10 times as great as 60,000.


6,000_____ is 10 times as great as 60,000.
The land area of _____Alaska__ is about 600,000 square miles.
So, __Alaska____ is about 10 times larger than Georgia.

Show and Grow

Use the table above.
Question 11.
Which state is about 10 times larger than Hawaii?
Answer: Georgia is 10 times larger than Hawaii

Question 12.
Which state is about \(\frac{1}{10}\) the size of Wyoming?
Answer: Maryland is 1/10 the size of Wyoming

Question 13.
DIG DEEPER!
Which state is about 100 times larger than the District of Columbia?
Answer: Hawaii

District of Columbia = 60

100 times larger than the District of Columbia 60 X 100 = 6,000 is Hawaii

Question 14.
DIG DEEPER!
A mother rhinoceros weighs 2 tons. Her baby weighs \(\frac{1}{10}\) as much as her. What is the weight of the baby rhinoceros, in pounds.

Answer: The weight of the baby rhinoceros is 20 Pounds.

2 tons Mother

Baby = 200 x 1/10 = 20 Pounds

Place Value Patterns Homework & Practice 1.1

Complete the statements.

Question 1.
______ is 10 times as great as 2,000.
Answer:   20,000

2,000 x 10 = 20,000

Question 2.
_____ is \(\frac{1}{10}\) of 2,000.
Answer:      200

2,000 x 1/ 10 = 200

Use a place value chart to answer the question.
Question 3.
What number is 10 times as great as 50?
Answer:  500

50 x 10 = 500

Question 4.
What number is \(\frac{1}{10}\) of 4,000?
Answer:  400

4,000 x 1/10 = 400

Question 5.
800 is \(\frac{1}{10}\) of what number?
Answer:8,000

8,000 x 1/10 = 800

Question 6.
60,000 is 10 times as great as what number?
Answer:6,000

6,000 is 10 times means 6,000 x 10 =60,000

Complete the table.
Question 7.

Answer:

1. 400, 4

40, 10 times as number is 40 x 10 = 400

1/10 of 40 is 4

2. 5,000, 50

500, 10 times as number is 500 x 10 = 5,000

1/10 of 500 is 50

3. 10,000  ,  100

1,000, 10 times as number is 1,000 x 10 = 10,000

1/10 of 1,000 is 100

4.8,00,000   , 8,000

80,000, 10 times as number is 80,000 x 10 = 8,00,000

1/10 of 80,000 is 8,000

Question 8.

Answer:

1. 7,00,000   ,   7,000

70,000, 10 times as number is 70,000 x 10 = 7,00,000

1/10 of 70,000 is 7,000

2. 200 ,  2

20, 10 times as number is 20 x 10 = 200

1/10 of 20 is 2

3. 30,000, 300

3,000, 10 times as number is 3,000 x 10 = 30,000

1/10 of 3000 is 300

4. 1,000 ,  10

100, 10 times as number is 100 x 10 = 1,000

1/10 of 100 is 10

Question 9.
Patterns
Describe the relationship between any place value position and the next lesser place value position.

Answer:

The relation between any place value position and next lesser place value position decreases ten times as we move.

Question 10.
YOU BE THE TEACHER
Your friend says 6,700 is \(\frac{1}{10}\) of 67,000. Is your friend correct? Explain.
Answer: Yes My friend is correct, because if we divide 67,000 by 10 we get the result as 6,700 only.

Use the table.

Question 11.
Modeling Real Life
Which city’s population is about 10 times the population of Newark?
Answer: Population of  Oakland.

Newark=40,000

40,000 x 10 = 4,00,000 is Oakland

Question 12.
Modeling Real Life
Which city’s population is about \(\frac{1}{10}\) the population of Marina?
Answer: Population of Del Ray Oaks.

Marina = 20,000

20,000 x 1/10 = 2,000=Del Ray Oaks

Question 13.
DIG DEEPER!
An archaeologist finds a ceramic bowl that is about 400,000 years old. He finds different artifact that is \(\frac{1}{100}\) times as old as the 100ceramic bowl. How much older is the ceramic bowl than the other artifact?
Answer: The Ceramic bowl is 4000 years old.

4,00,000 x 1/100= 4,000

Review & Refresh

Find the factor pairs for the number.
Question 14.
9
Answer: 1,3,9 are factor pairs of 9

Question 15.
24
Answer:  (1,24), (2,12) (3,8) and (4,6) are factor pairs of 24

Question 16.
15
Answer:(1,3,5,15) are factor pairs of 15

Lesson 1.2 Place Value with Whole Numbers

Model the number. Draw your model.
Then write the value of each digit.

Compare the values of the 4s.
Answer: Thousands-4, Hundreds -4 , Tens – 4 and Ones- 2

4 is in thousand, 4 is in Hundreds and 4 is at tens value.

Repeated Reasoning
Is the value of the 4 in the tens place 10 times as much as the value of the 2 in the ones place? Explain.
Answer: No, Why means 4 in tens place means its value is 40 and 2 in ones place means only 2so 4 in the tens place is not 10 times as the value of 2.

Think and Grow: Place Value with Whole Numbers

Key Idea
A place value chart shows the value of each digit in a number. It also shows how the place values are grouped. Each group of three digits is called a period. Ina number, periods are separated by commas.

Example
Write the number in standard form, word form, and expanded form.
Standard form:4,66,900
Word form: Four hundreds sixty six thousand and nine hundred
Expanded form:
4 × 1,00,000 + __6___ × 10,000_____ + ___6__ × _1,000_____ + _9____ × __100____

Show and Grow

Write the number in two other forms.
Question 1.
Standard form: 78,300
Word form:
Expanded form:
Answer:

Word Form : Seventy -Eight Thousand, Three Hundred.

Expanded Form : 7 x 10,000 + 8 x 1000 + 3 x 100=78,300

Question 2.
Standard form:
Word form: three hundred fifty thousand, fifty-eight
Expanded form:
Answer:

Standard Form : 300,50,058

Expanded Form : 3 x 10,00,000+ 5 x 10,000 + 5 x 10 +8=300,50,058

Question 3.
Compare the values of the 6s in the number 466,900.
Answer:

the values of the 6 s are one is in 60 thousand’s place and other is at 6 thousands place.

Apply and Grow: Practice

Write the number in two other forms.
Question 4.
Standard form:
Word form:
Expanded form: 6 × 100,000 + 8 × 1,000 + 4 × 100 + 5 × 10 + 9 × 1
Answer:

Standard Form : 6,08,459

Word Form : 6 hundred / six lakh, eight thousand , four hundred  fifty nine.

Question 5.
Standard form: 45,006,702
Word form:
Expanded form:

Answer:

Word form: forty five lakh ,six thousand seven hundred and two.

Expanded form:4 x 10000000 + 5 x 1000000 + 6 x 1000 +

7 x 100 +2

Question 6.
Compare the values of the 7s in the number 4,877,034.
Answer:

The value’s of 7s is at seventy thousand,(70,000) and again at  seven thousand(7,000).

Question 7.
Compare the values of the 3s in the number 5,338.
Answer:

The values of 3s is at 3-Hundred(300) and at thirty (30)[3 tens)]

or

3 at hundreds and 3 at tens place.

Compare.
Question 8.

Answer:

In 8,046 The value of 4 is in Ten’s place and 6 is in one’ s place

and in 8,460 the value of 4 is in hundreds place and 6 in ten’s place.

Question 9.

Answer:

In 28,517 the value of 2 is at twenty thousand place, 8 at thousands and 5 at hundreds, 1 at tens and 7 at one’s place.

and 28,509 the value of 2 is at twenty thousand place, 8 at thousands and 5 at hundreds, zero at tens and 9 at one’s place.

Question 10.

Answer:

In 5,854,331- Fifty Lakhs , 8 at lakhs, Five at ten thousand, 4 at thousand,3 at hundred, 3 at ten’s and 1 at one’s place.

and in 5,854,231 – Fifty Lakhs , 8 at lakhs, Five at ten thousand, 4 at thousand,2 at hundred, 3 at ten’s and 1 at one’s place.

Question 11.
The white truffle is the world’s most expensive edible fungus, which costs up to three thousand dollars per kilogram. Write this number in standard form.

Answer:

Standard Form: 3000$ per Kg.

the white truffle is 3000$ per kg.

Question 12.
YOU BE THE TEACHER
Your friend says that in the number 45,951, one 5 is 10 times as great as the other 5. Is your friend correct? Explain.
Answer:

Yes,  because  at first the 5 is at tens place and in next time 5 is at thousands place so friend is right 5 is 10 times greater as the other 5 . As we move from right value to left twice tens value place becomes thousand value place.

Question 13.
Logic
Newton is thinking of a 6-digit number in which all of the digits are the same. The value of the digit in the thousands place is 8,000. What is Newton’s number?
Answer:

8,88,888

8 x 1,00,000 + 8 x 10,000+8 x 1,000+8 x 100 + 8 x 10 + 8 x 1

Think and Grow: Modeling Real Life

Example
Compare the values of the 3sin Jupiter’s average distance from the Sun.

Use a place value chart to help you find the value of each 3.

Each place value is 10 times as great as the place value to its right. The digits are two places apart. So, multiply 30,000 by 10 × 10 = 100.
So, the value of the 3 in the millions place is _______ times the value of the 3 in the ten thousands place.

Show and Grow

Use the table above.
Question 14.
Compare the values of the 7s in Mars’s average distance from the Sun.
Answer:

the value of 7 s in first is in thousands place and next 7 s is at hundreds place, the digits are one place apart, so the value of first 7 is in thousands place is 10 times the value of the next 7s in hundreds place.

Question 15.
Compare the values of the 4s in Saturn’s average distance from the Sun.
Answer:

the value of 4 s in first is in four hundred thousands place and next 4 s is at hundreds place, the digits are two places apart, so the value of first  4 is in hundred thousands place is 100 times the value of the next 4s in hundreds place.

Question 16.
DIG DEEPER!
An organization wants to donate all of the money raised through fund raisers and raffles to a children’s charity. Complete the donation check.


Answer:

1.

$ 194,918 + $ 35,187 = $ 230,005

Word Form :  Two Hundred, Thirty thousand and Five dollars.

Standard Form : $ 230,005

Place Value with Whole Numbers Homework & Practice 1.2

Write the value of the underlined digit.
Question 1.
740,225
Answer:

4 is at Forty thousand place

Question 2.
604,197,872
Answer:

6 is at Six Hundred Lakh place

Question 3.
12,405,287
Answer:

2  is at  twenty Lakh or twenty hundred thousand place

Question 4.
392,183
Answer:

3 is at 3 lakhs or 3 hundred thousand place

9  is at ninety thousand place

2 is at 2 thousand place

1 is at one hundred place

8 is at eighty place

3 is at ones place place

Write the number in two other forms.
Question 5.
Standard form: 450,014
Word form:
Expanded form:
Answer:

Word Form : 4 lakhs or 4 hundred thousand ,  fifty thousand and fourteen

Expanded  Form : 4 x 1,00,000 +5 x 10000 + 1 x 10 + 4

Question 6.
Standard form:
Word form: fourteen thousand, two hundred one
Expanded form:
Answer:

Standard Form : 14,201

Expanded Form : 1 X 10000 + 4 X 1000 + 2 X 100 +1

Question 7.
Compare the values of the 9s in the number 537,499.
Answer:

First 9 is at ones place, and second 9 is at tens place.

Question 8.
Compare the values of the 5s in the number 78,550,634.
Answer:

First 5 is at Ten Thousands place,

Second 5 is at Five Hundred Thousands place.

Compare.
Question 9.

Answer:

67,893 < 67,943

6 at sixty thousand place ,6 at sixty thousand place

7 at thousand place, 7 at thousand place

8 at eight hundred place,9 at nine hundred is great

9 at tens place and 4 at four at tens place is less

and 3 at ones place and 3 at ones place is same

Question 10.

Answer:

450,823 > 405,823

4 at Four hundred thousand,

5 at fifty thousand and 0 is smaller at ten thousands place

0 at thousands place is smaller than 5 at thousands place

8  at hundreds place is same as 8 at hundreds place

2 at tens place is same as 2 at tens place

and 3 at ones place is same as 3 at ones place

Question 11.

Answer:

176,994 = 176,994

1 at one hundred thousand, is same at 1 at One Hundred Thousand place

7 at Seventy thousand is same at 7 at Seventy Thousand place

6 at six thousand is same at 6 at six thousand place

9 at hundreds place is same at 9 at hundreds place

9 at ninety or 9 tens place is same as 9 at Ninety or 9 Tens place

and 4 at ones place is same as 4 at Ones place

Question 12.

Your body contains about 60,000 miles of blood vessels. Write this number in word form.
Answer:

60,000 miles of blood vessels in  Word Form : Sixty Thousand miles.

Question 13.
Which One Doesn’t Belong?
Which number does belong with the other three?
1 × 10,000 + 4 × 1,000 + 2 × 100 + 6 × 1 fourteen thousand, two hundred six
140,206

Answer:

No number is repeated, so no number belongs to other

one at tens place

4 at ones place

2 at hundredths place

and 6 at thousandths place.

Question 14.
DIG DEEPER!
Find the difference in the values of the underlined digits.
856,092 37,841
Answer:

8 is at 80 Million,

and

8 is at hundreds value

the difference is 8 X 1,00,00,000 times the other  8 value.

Use the table.

Question 15.
Modeling Real Life
Compare the values of the 3s in the music cost.
Answer: 3 is at 3 millions place and other 3 is at 3 hundreds place.

Question 16.
Modeling Real Life
Compare the values of the 2s in the cost.
Answer:

Costs Places

In cast 2 is at two hundred thousands place and 2 at thousands place.

In Director 2 is at 2 millions place and 2 at thousands place.

In Editing 2 value is not there.

In Music also 2 is not there

In Producers 2 value is at 2 millions place

In Visual effects 2 place is first at thousands place and 2 is in ones place

Question 17.
DIG DEEPER!
What is the total cost for the director and producers? Write your answer in word form.
Answer: Total cost of director-$ 2,712,480+ and cost of roducer –

$ 2,759,981 is $ 5,472,461

Word Form : five million, four hundred thousand, seventy two thousand, four hundred sixty one dollars.

Review & Refresh

Compare
Question 18.

Answer:

0.14 < 0.15

0.14-0+ 1 x 1/10+4/100

0 is at ones place ,1 is at 1/10 place, and 4 is at 1 /100 place is small

0.15-0+1×1/10+5/100

0 is at ones place ,1 is at 1/10 place, and 5 is at 1 / 100 place

Question 19.

Answer:

2.2 = 2.20

2.2-2 is at ones place and .2 is at 2/10 place

2.20- 2 is at ones place ,.2 is at 2/10 place is same

Question 20.

Answer:

5.8 > 5.08

5.8 – 5 is at ones place and .8 is at 1/10 place

5.08  – 5 is at ones place and .8 is at 1/100 place  is small

Lesson 1.3 Patterns and Powers of 10

Explore and Grow

Write a multiplication expression to answer each question.

How many units are in 1 rod?

How many units are in 10 rods?

How many units are in 100 rods?

How many units are in 1,000 rods?
Answer:

In 1 rod its 1 x 10^{0 =}10⁰

In 10 rods its 1 x 10 = 10¹

In 100 rods its 1 x 10 x 10 = 10²

In 1,000 rods it is 1 x 10 x 10 x 10 =10³

Repeated Reasoning
How many tens are in 100? in 1,000? in 10,000?
Answer: In 100 its 10 tens, in 1,000 its 100 tens and in 10,000 its 1000 tens

Think and Grow: Patterns and Powers of 10

Key Idea
A power is a product of repeated factors. The base of a power is the repeated factor. The exponent of a power gives the number of times the base is used as a factor.

Example
Find the value of 4 × 10³.

Multiply 4 by powers of 10. Look for a pattern.
4 × 10¹ = 4 × 10 = _40____
4 × 10² = 4 × 10 × 10 = _400_____
4 × 10³ = 4 × 10 × 10 × 10 = __4,000____
So, 4 × 10³ = ____4,00,000_.
Notice the pattern: In each product, the number of zeros after 4 is equal to the exponent.

Show and Grow

Question 1.
Identify the base, exponent, and power for the expression 10⁶.
Answer:

base : 10,

exponent: 6

power :10⁶⁼ 10 x 10 x 10 x 10 x 10 x 10

Question 2.
Write the product 10 × 10 × 10 × 10 as a power.

Answer:

10 x 10 x 10 x 10  as a power is 10 ⁴

Question 3.
Find the value of 5 × 10².
Answer:

5 × 10^{2 is} 

5 x 10 x 10 = 500

Apply and Grow: Practice

Find each product. Use patterns to help.
Question 4.
2 × 10 = _____
2 × 100 = _____
2 × 1,000 = _____
2 × 10,000 = ____
Answer:

2 x 10 =20

2 x 100=200

2 x 1,000= 2,000

2 x 10,000 = 20,000

Question 5.
9 × 10 = _____
9 × 100 = _____
9 × 1,000 = _____
9 × 10,000 = ____
Answer:

9 x 10 = 90

9 x 100 = 900

9 x 1,000= 9,000

9 x 10,000 = 90,000

Question 6.
5 × 10 = _____
5 × 100 = _____
5 × 1,000 = _____
5 × 10,000 = ____
Answer:

5 x 10 = 50

5 x 100= 500

5 x 1,000 = 5,000

5 x 10,000 = 50,000

Find the value of the expression.
Question 7.
10⁴
Answer:

10⁴⁼ 10 x 10 x 10 x 10 = 10,000

Question 8.
6 × 10⁵
Answer:

6 × 10^{5 =} 6 x 10 x 10 x 10 x 10 x 10 = 6,00,000

Question 9.
7 × 10²
Answer:

7 × 10^{2 =} 7 x 10 x 10 = 700

Question 10.
5 ×10 ⁴
Answer:

5 ×10 ⁴⁼ 5 x 10 x 10 x 10 x 10 = 50,000

Rewrite the number as a whole number multiplied by a power of 10.
Question 11.
20,000
Answer:

20,000=2 x 10⁴

Question 12.
500
Answer:

500=5 X 10²

Question 13.
900,000
Answer:

9,00,000=9 x 10⁵

Number Sense
Write the number in expanded form using exponents.
Question 14.
53,124
(5 × 10⁴) + ______
Answer:

53,214=(5 × 10⁴) +( 3 x 10³)+(1 x 10²) + (2 x10¹)+ (4 x10⁰)

Question 15.
8624
(8 × 10²) + _______
Answer:

8624=(8 x 10³ )+(6 x 10²) +(2 x 10¹)+(4×10⁰)

Question 16.
DIG DEEPER!
Your friend writes (3 × 10⁴) + (5 × 10³) + (2 × 10²) + 4 as the expanded form of thirty-five thousand, twenty-four. Explain what your friend did wrong.
Answer:

(3 × 10⁴) is 3 x 10 x 10 x 10 x 10 = 30,000 is Thirty Thousand

(5 × 10³) is 5 x 10 x 10 x 10 = 5,000 is five thousand

(2 × 10²) is 2 x 10 x 10 = 200 is two hundred not twenty

and 4 is four,

it is  thirty-five thousand, two hundred and four= 35,204

not thirty-five thousand, twenty-four ≠ 35,024

35,204 is right

35,024 is wrong

Think and Grow: Modeling Real Life

Example
Newton and Descartes are running for mayor. How many people voted in the election?
Find the number of votes for each candidate.
Newton: 10⁵ = ______
Descartes: 9 × 10⁴ = ______
Add the votes for Newton and Descartes.

______ people voted in the election.

Newton: 10⁵ =  10 x 10 x 10 x 10 x 10 is 1,00,000

Descartes: 9 × 10⁴ = 9 x 10 x 10 x 10 x 10 is 90,000

so total people voted is 1,00,00 + 90,000 = 1,90,000

1,90,000 Voted in the election.

Show and Grow

Question 17.
A surf shop has been in business for two years. What are the total sales for Year 1 and Year 2 combined?

Answer:

Year 1 $ 10^{4 =} 10 x 10 x 10 x 10 =10,000

Year 2 $ 6 x 10⁵= 6 x 10 x 10 x 10 x 10 x 10 = 6,00,000

there fore total sales for Year 1 and Year 2 combined is  is 6,10,000 .

Question 18.
Which migration is farther? About how much farther is it?

Answer:

Chinook Salmon about 4 x 10^3km

4 X 10 x 10 x 10 = 4000 km

and

Leatherback Turtle about 2 X 10⁴ km

2 x 10 x 10 x 10 x 10 = 20,000 km

So Leatherback Turtle is  farther and

20,000 km – 4000 km = 16,000 km

it is 16,000 km farther

Question 9.
DIG DEEPER!
A human has about 10⁴ taste buds. A cow has about 3 times as many taste buds as a human. About how many taste buds does a cow have? Write your answer as a whole number multiplied by a power of 10.
Answer:

(10⁴) 3 time means (10⁴)³ 

as per the law powers are multiplied 4 x 3

(10^4×3) = (10¹²)

(10^4×3)= 1 X (10⁴) x (10⁴)  x (10⁴)

(10¹²)= 1 x (10 x 10 x 10 x 10 ) x (10 x 10 x 10 x 10 ) x (10  x 10 x 10 x 10)

Patterns and Powers of 10 Homework & Practice 1.3

Question 1.
Identify the base, exponent, and power for the expression 10³.
Answer:

base: 10

exponent:  3

power : 10³⁼ 10 x 10 x 10

Question 2.
Write 10 × 10 × 10 × 10 a power.
Answer:

10 x 10 x 10 x 10 = 10⁴

Find each product. Use patterns to help.
Question 3.
6 × 10 = ______
6 × 100 = _____
6 × 1,000 = _____
6 × 10,000 = _____
Answer:

6 x 10=6 × 10^{1 =} 60

6 x 100 =6 × 10² = 600

6 X 1,000=6 × 10^{3 =}  6,000

6 x 10,000 = 6 × 10⁴= 60,000

Question 4.
8 × 10 = ______
8 × 100 = _____
8 × 1,000 = _____
8 × 10,000 = _____
Answer:

8 x 10=8 × 10^{1 =} 80

8 x 100 =8 × 10² = 800

8 X 1,000=8 × 10^{3 =} 8,000

8 x 10,000 = 8 × 10⁴= 80,000

Question 5.
4 × 10 = ______
4 × 100 = _____
4 × 1,000 = _____
4 × 10,000 = _____
Answer:

4 x 10=4 × 10^{1 =} 40

4 x 100 =4 × 10² = 400

4 X 1,000= 4× 10^{3 =} 4,000

4 x 10,000 = 4 × 10⁴= 40,000

Find the value of the expression.
Question 6.
10³
Answer:

10^{3 =} 10 x 10 x 10 = 1,000

Question 7.
2 × 10⁴
Answer:

2 × 10^{4 =} 2 x 10 x 10 x 10 x 10 = 20,000

Question 8.
9 × 10⁵
Answer:

9 × 10^{5   =}  9 x 10 x 10 x 10 x 10 x 10 = 9,00,000

Question 9.
3 × 10²
Answer:

3 × 10^{2  =} 3 x 10 x 10 = 300

Rewrite the number as a whole number multiplied by a power of 10.
Question 10.
100,000
Answer:

100,000 = 1 x 10 x 10 x 10 x 10 x 10 = 1 × 10⁵

Question 11.
70
Answer:

70 = 7 x 10 = 7 x 10¹

Question 12.
6,000
Answer:

6,000 = 6 x 10 x 10 x 10 = 6 x 10³ 

Number Sense
Write the number in standard form.
Question 13.
(3 × 10²) + (8 × 10¹)
Answer:

(3 × 10²) + (8 × 10¹)  = 3 x 10 x 10 + 8 x 10

3 x 100 + 80

300+ 80 = 380

Question 14.
(2 × 10³) + (5 × 10²) + (4 × 10¹)
Answer:

(2 × 10³) + (5 × 10²) + (4 × 10¹) =  2 x 10 x 10 x 10 + 5 x 10 x 10 + 4 x 10

2,000 + 500 + 40

2,540

Question 15.
YOU BE THE TEACHER
Newton 6 says 10⁶ = 10 × 6. Is he correct? Explain
Answer:

No he is wrong because  10⁶  ≠  10 × 6 it is 10 is to be multiplied by  6 times ,

10^{6 =} 10 x 10 x 10 x 10 x 10 x 10=10,00,000 is correct

Question 16.
Which One Doesn’t Belong?
Which one does not belong with the other three?

Answer:

Both are different blocks , in first block it is  square and next it is cube.  

Question 17.
Modeling Real Life
Each student at an elementary school votes once on this year’s field trip. How many students voted in all?

Answer:

Aquarium = 7 x 10² =   7 x 10 x 10 = 700

Amusement Park = 10^{3  =} 10 x 10 x 10 =  1000

700 + 1000 = 1,700

So total number of students voted are =  1,700.

Question 18.
Modeling Real Life
On which day did more people attend the event? How many more people?

Answer:

Friday = 10 x 10 x 10 = 1,000

Saturday = 5 x 10² = 5 x 10 x 10 = 500

on Friday more people attend the event

and more are 1,000 – 500 = 500 , s0 500 more people attended.

Review & Refresh

Divide. Then check your answer
Question 19.

Answer:

Step 1) Start by dividing 65 by 6 to get the decimal answer as illustrated below. Note that we round the answer if necessary, but don’t worry, the final answer will still be exact.

65 / 6 = 10.83

Step 2) Next we take the Whole part of the answer in Step 1 and multiply it by the Divisor. As you can see, it does not matter if we rounded in the previous step because the Decimal part is ignored. Furthermore, the Divisor in 65 divided by 6 is 6. Thus, the Whole multiplied by the Divisor is:

10 x 6 = 60

Step 3) Finally, we will subtract the answer in Step 2 from the Dividend to get the answer. The Dividend in 65 divided by 6 is 65. Thus, our final calculation to get the answer is:

65 – 60 = 5

the answer is 5.

Question 20.

Answer:

Step 1) Start by dividing 50 by 4 to get the decimal answer as illustrated below. Note that we round the answer if necessary, but don’t worry, the final answer will still be exact.

50 / 4 = 12.50

Step 2) Next we take the Whole part of the answer in Step 1 and multiply it by the Divisor. As you can see, it does not matter if we rounded in the previous step because the Decimal part is ignored. Furthermore, the Divisor in 50 divided by 4 is 4. Thus, the Whole multiplied by the Divisor is:

12 x 4 = 48

Step 3) Finally, we will subtract the answer in Step 2 from the Dividend to get the answer. The Dividend in 50 divided by 4 is 50. Thus, our final calculation to get the answer is:

50 – 48 = 2

the answer is 2.

Question 21.

Answer:

Step 1) Start by dividing 45 by 3 to get the decimal answer as illustrated below. Note that we round the answer if necessary, but don’t worry, the final answer will still be exact.

45 / 3 = 15

the answer is 0.

Lesson 1.4 Decimals to Thousandths

Explore and Grow

Divide the square into10 equal parts. Shade one part. What part of the whole is shaded?
Fraction: Decimal:

Divide each of the 10 parts into 10 equal parts. Shade one part using a different color. What part of the whole is shaded with the second color?

Fraction: Decimal:

If you divide each of the 100 equal parts into10 equal parts, how many parts will the model have?

If you shade one of those parts, what part of the whole is shaded?
Fraction: Decimal:
Answer:

the model will have 10 parts , only 1/ 10th part is shaded, 0.1

Structure
Compare the number of hundredths to the number of tenths. Compare the number of hundredths to the number of thousandths. What do you notice?
Answer:

number of hundredths to the number of tenths is 10 to 100 , 10/100=1/10 = 0.1

number of hundredths to the number of thousandths is

100 to 1000, 100/1000 =1/10= 0.1

both has equal values 1/10 = 0.1

the number of hundredths to the number of tenths is equal to the number of hundredths to the number of thousandths

Think and Grow: Thousandths

Key Idea
In a decimal, the third place to the right of the decimal point is the thousandths place. You can write thousandths as fractions or decimals.

Show and Grow

Write the decimal as a fraction.
Question 1.
0.009

Answer:

0.009= 9 / 1,000 = 9 x 1/1,000

Question 2.
0.063
Answer:

0.063 = 63 / 1,000= 63 x 1/1,000

Question 3.
0.194
Answer:

0.194 = 194 / 1,000= 194 x 1/1,000

Write the fraction as a decimal
Question 4.
\(\frac{3}{1,000}\)
Answer:

\(\frac{3}{1,000}\)= 3 x 1/ 1,000=  0.003

Question 5.
\(\frac{91}{1,000}\)
Answer:

\(\frac{91}{1,000}\)= 91 x 1/ 1,000 = 0.091

Question 6.
\(\frac{607}{1,000}\)
Answer:

\(\frac{607}{1,000}\)= 607 x 1/1,000= 0.607

Apply and Grow: Practice

Write the decimal as a fraction.
Question 7.
0.645
Answer:

0.645= 645x 1/1,000= 645/1,000

Question 8.
0.002
Answer:

0.002=2 x 1/1,000=  2/1,000

Question 9.
0.98
Answer:

0.98= 98 x 1/ 1,000= 98/1,000

Question 10.
0.6
Answer:

0.6 = 6 x 1/10= 0.6/10

Write the fraction as a decimal.
Question 11.
\(\frac{884}{1,000}\)
Answer:

\(\frac{884}{1,000}\)= 884x 1/1,000= 0.884

Question 12.
\(\frac{8}{1,000}\)
Answer:

\(\frac{8}{1,000}\)= 8 x 1/1,000= 0.008

Question 13.
\(\frac{39}{100}\)
Answer:

\(\frac{39}{100}\)= 39 x 1/ 100= 0.39

Question 14.
\(\frac{1}{10}\)
Answer:

\(\frac{1}{10}\)= 1 x 1/10 = 0.1

Question 15.
0.4 is \(\frac{1}{10}\) of what number?
Answer:

\(\frac{4}{10}\) is 0.4

Question 16.
0.52 is 10 times as great as what number?
Answer:

0.52 is 10 times as great as 0.052

Question 17.
You use 47 of the cotton balls for an art project. What portion of the bag of cotton balls do you use? Write your answer as a decimal.

Answer:

47/100= 47 x 1/100= 0.47

Question 18.
Which One Doesn’t Belong?
Which number does not belong with the other three?

Answer:

0.29 does not belongs to other three

Question 19.
YOU BE THE TEACHER
Your friend says the value of the 7 in the hundredths place of 0.877 is 10 times as great as the 7 in the thousandths place. Is your friend correct? Explain.
Answer:

Yes , because the value of 7 in the hundredths place as compared is 10 times as great as the 7 in the thousands place.

Question 20.
Write each fraction as a decimal. What do you notice?
\(\frac{4}{10}\), \(\frac{40}{100}\) and \(\frac{400}{1,000}\)
Answer:

\(\frac{4}{10}\)=0.4

\(\frac{40}{100}\) =0.4

\(\frac{400}{1,000}\)=0.4

number of tenths, tenths number of hundredths and hundredths number of thousandths are same.

Think and Grow: Modeling Real Life

Example
You put together 156 pieces of the puzzle before lunch and 148 pieces of the puzzle after lunch. What portion of the puzzle did you put together? Write your answer as a decimal.


Write the fraction as a decimal.
You put together ____0.304__ of the puzzle.

Show and Grow

Question 21.
You make flash cards out of index cards. You use 50 index cards for social studies and 25 index cards for science. What portion of the pack of index cards do you use? Write your answer as a decimal.

Answer:

50 for social studies, 25 for science total is 50 + 25 = 75, total number of flash cards is 1,000 ,

75 / 1,000 or 75 by 1,000 or 75 x 1/1,000= 0.075

Question 22.
There are 458 knock-knock jokes in the book. not What fraction of the jokes in the book are knock-knock jokes?

Answer:

458/1,000 or 458 x 1/1,000 of jokes books in knock- knock jokes.

Question 23.
DIG DEEPER!
A newly hatched caterpillar was 0.02 inches long. After 2 weeks, the caterpillar grew 10 times as long as its length when it hatched. After another 2 weeks, the caterpillar grew 10 times as long as its length after 2 weeks. How long is the caterpillar now?
Answer: First week it is 0.02 inches,

in two weeks -2 weeks-0.02 x 10 = 0.2 inches

again after 2 weeks -0.2 x 10 = 2 inches

Decimals to Thousandths Homework & Practice 1.4

Write the decimal as a fraction.
Question 1.
0.735
Answer:

0.735= 735 / 1,000= 735 x 1/1,000

Question 2.
0.051
Answer:

0.051= 51 / 1,000= 51 x 1/1,000

Question 3.
0.804
Answer:

0.804 = 804 / 1,000= 804 x 1/ 1,000

Question 4.
0.2
Answer:

0.2 = 2 / 10= 2 x 1/10

Write the fraction as a decimal.
Question 5.
\(\frac{98}{1,000}\)
Answer:

\(\frac{98}{1,000}\)=98 x 1/1,000= 0.098

Question 6.
\(\frac{67}{100}\)
Answer:

\(\frac{67}{100}\)= 67 x 1/100=0.67

Question 7.
\(\frac{4}{100}\)
Answer:

\(\frac{4}{100}\)= 4 x 1/100=0.04

Question 8.
\(\frac{9}{10}\)
Answer:

\(\frac{9}{10}\)= 9 x 1/10= 0.9

Question 9.
0.08 is 10 times as great as what number?
Answer:

0.008

if 0.008 is multiplied by 10 times it becomes 0.08

Question 10.
0.001 is \(\frac{1}{10}\) of what number?
Answer:

0.0001

if 0.0001 is \(\frac{1}{10}\) equals to 0.001

Question 11.
YOU BE THE TEACHER
Your friend says that \(\frac{16}{1,000}\) can be written as 0.16. Is your friend correct? Explain.
Answer: No, because if 16 / 1,000 or 16 x 1/1,000 is 0.016 not 0.16,

0.016 ≠ 0.16

so he is incorrect.

Question 12.
Precision
Thirteen unit cubes are taken from the thousand cube. Write a fraction and a decimal to represent how many unit cubes are left.

Answer:

13 units cubes are taken

Total number of units are  1000

left are from 1,000-13/1,000=87/1000

87 / 1,000 and 0.087 cubes are left

Question 13.
DIG DEEPER!
Write the number represented by each point on the number line.

Point X: _____
Point Y: ___
Answer:

on the number line Point X = 7.633 as we move from 7.63 to three places forward

so Point X is 7.633

on the number line Point Y = 7.638 as we move from 7.63 its eighth place forward

and Point Y is 7.638

Question 14.
Modeling Real Life
A restaurant owner has a 1,000-count box of napkins. She puts 125 of the napkins on tables. What portion of the box of napkins does she use for the tables? Write your answer as a decimal.
Answer:

Restaurant has 1,000 count box of napkins and keeps 125 on table so portion of the box she uses is  125/1,000= 125 x 1/ 1,000= 0.125

Question 15.
DIG DEEPER!
Your friend has a recipe book with 1,000 recipes. She wants to try two new recipes each week. What fraction of the recipes in the book will she have tried after 1 year?
Answer:

In a year there are almost 52 weeks. Each week 2 means 2 x 52 =approximately  104 recipes in a year.

so in a year she would have tired 104 / 1,000= 104 x 1/1,000 = 0.104 recipes.

Review & Refresh

Question 16.
Extend the pattern of shapes by repeating the rule “square, octagon, pentagon, octagon ”What is the 48th shape in the pattern?

Answer:

” square, octagon, pentagon, octagon, hexagon, octagon , heptagon , octagon,

octagon, octagon, nonagon, octagon , decagon, octagon,

Lesson 1.5 Place Value with Decimals

Explore and Grow

Model the number. Draw your model.
Then write the value of each digit.

Answer:

from the model 3.33 =

0nes digit is – 3

tenths digit is 3/10 =0.3

hundredths digit is 3/ 100 = 0.03

Repeated Reasoning
Compare the value of the ones digit to the value of the tenths digit. Then do the same with the tenths and the hundredths digits. Explain why you can use base ten blocks to model ones, tenths, and hundredths.
Answer:

a digit in one place represents one and 10 times more what it represents in the place to its right and that is tenths digit.
Similarly a digit in tenths place represents tenth and 10 times more what it represents in the place to its right and that is hundredths digit.

Each place to the left is 10 times the size of the place to the right, and base 10 blocks are the best way to model ones, tenths, and hundredths.

Think and Grow: Place Value with Decimals

Key Idea
In a place value chart, whole numbers are to the left of the decimal point. Decimals are to the right of the decimal point.

Standard form:2.557
Word form:” Two and five hundred fifty-seven  thousandths”
Expanded form: 2 × 1 + __5__ ×\(\frac{1}{10}\)____ + 5 × \(\frac{1}{100}\) + _7_ × \(\frac{1}{1000}\)__

Show and Grow

Write the number in two other forms.
Question 1.
Standard form: 0.398
Word form:
Expanded form:

Answer:

Word form: “Three hundred ninety – eight thousandths”
Expanded form: 3 x (1/10) + 9 x (1/100) + 8 x (1/1,000)

Question 2.
Standard form:
Word form: eight and forty-six thousandths
Expanded form:
Answer:

Standard form:8.046

Expanded form: 8 x 1 + (4 /100) + (6 / 1,000)

Question 3.
Compare the values of the 5s in the number 2.557.
Answer: at first 5s place value is at tenths and next its place value is at hundredths .

Apply and Grow: Practice

Write the value of the underlined digit.
Question 4.
0.418
Answer:

4 is at tenths value place

Question 5.
5.296
Answer:

9 is at hundredths value place

Question 6.
3.806
Answer:

8 is at tenths value place

6 is at  thousandths value place

Question 7.
0.547
Answer:

7 is at thousandths value place

Write the number in two other forms.
Question 8.
Standard form:
Word form:
Expanded form: 4 × 1 + 9 × \(\frac{1}{10}\) + 8 × \(\frac{1}{1,000}\)
Answer:

Standard form: 4.908
Word form:” Four and nine hundredth – eight thousandths”

Question 9.
Standard form: 0.125
Word form:
Expanded form:
Answer:

Word form: “one hundred twenty – five thousandths”
Expanded form: 1 x (1/10) + 2 x (1 /100) +5 x (1 / 1,000)

Question 10.
Compare the values of the 4s in the number 0.844.
Answer:

at first 4s place value is at hundredths and next 4s place value is at thousandths

Question 11.
Compare the values of the 3s in the number 3.367.
Answer: at first 3s place value is at ones place and next 3s place value is at tenths place

Question 12.
A pygmy jerboa weighs one hundred thirty-two thousandths pound. Write this number in standard form.

Answer:

The Standard Form of pygmy jerboa’s -one hundred thirty-two thousandths pound weighs is  0.132 pound

Question 13.
Reasoning
Is 9.540 equivalent to 9.54? Explain.
Answer:

Yes. 9.540 is equivalent to 9.54 because at thousandths value its 0, so zero multiplied by any number is zero.

Therefore both are equivalent.

Question 14.
DIG DEEPER!
Write three decimals that are equivalent to 6 × 1 + 4 × \(\frac{1}{10}\) .
Answer:

the three equivalent decimals are of 6 .04 are 6.040, 6.0400, 6.04000

Think and Grow: Modeling Real Life

Example
How do the values of the 3s in the masses of the fruits compare?

Use a place value chart to help you find the value of each 3.

The value of the 3 in the mass of the tomato is _one_____ .
The value of the 3 in the mass of the chili pepper is __tenths___.
So, the value of the 3 in the mass of the tomato is __10_____ times the value of the 3 in the mass of the chili pepper. Also, the value of the 3 in the mass of the chili pepper is ___1/10__ the value of the 3 in the mass of the tomato.

Show and Grow

Question 15.
Two baseball players have batting averages of 0.358 and 0.345. How do the values of the 5s in the batting averages compare?
Answer:

In 0.358 the place value of 5s is at tenths place and in

0.345 the place value of 5s  is at thousandths place.

Question 16.
The stopwatch shows a runner’sritethe100-meter dash time. Write the time in words.

Answer:

15.76 seconds time in words  is Fifteen and seven tenths and six hundredths

Question 17.
DIG DEEPER!
You exchange 1 U.S. dollar for Australian dollars and 1 U.S. dollar for Kuwaiti dinars. Do you have 10 times as many Australian dollars as Kuwaiti dinars? Explain.

Answer:

Yes, Because 1 Australian dollars = 1.302 of 0.302 of Kuwaiti dinars,

if we multiply 0.302 by 10 it becomes 1.302 which is equal to 1 Australian dollars.

[0.302 x 10 = 1.302]

so 1 Australian dollars is 10 times more than the 1  Kuwaiti dinars.

Place Value with Decimals Homework & Practice 1.5

Write the value of the underlined digit.
Question 1.
5.437
Answer:

5 at ones place

4 at tenths place

3 at hundredths place

7 at thousandths place

Question 2.
0.852
Answer:

the underlined digit is 2 at thousandths place,

its value is 2 x 1/1000= 2/1000

Question 3.
0.962
Answer:

the underlined digit is 6 at hundredths place

its value is 6 x 1/100= 6 /100

Question 4.
4.165

the underlined digit is 1 at tenths place

its value is 1x 1/10 = 1/10
Answer:

Write the number in two other forms.
Question 5.
Standard form: 9.267
Word form:
Expanded form:
Answer:

Word form:  nine and two tenths six hundredths seven thousandths.

Expanded form : 9+ 2 x 1/10 + 6 x 1/100 + 7 x 1/1000

Question 6.
Standard form:
Word form: two and forty-three thousandths
Expanded form:
Answer:

Standard form : 0.243

Expanded form: 2 x 1/10+4 x 1/100+3 x 1/1000

Question 7.
Compare the values of the 6s in the number 1.668.
Answer:

first 6s at tenths place and next 6s at hundredths place

Question 8.
Compare the values of the 7s in the number 7.704.
Answer:

first 7s at ones place and next 7s at tenths place

Question 9.
A pygmy possum weighs 0.097 pound. Write this number in word form.

Answer:

pygmy weighs 0.097 pound and its

word form is  nine hundredths and seven thousandths

Questio 10.
Which One Doesn’t Belong?
Which one does not belong with the other three?

Answer:

no number belongs to other three because 5 is at tenths place

one at hundredths place and 4 at thousandths all are at different places.

Question 11.
Reasoning
Which number cards are equal to the value of the underlined digit?

Answer:

2 x 1/1,000 , two thousandths and 0.002 are equal to the value of the underlined digit.

Question 12.
Modeling Real Life
How do the values of the 5s in the heights of the plants compare?

Answer:

In Peace lily the 5s place is at  thousandths

and in Venus flytrap the 5s place is at hundredths

Question 13.
Modeling Real Life
The world’s largest gold nugget is located in Las Vegas, Nevada. It has a mass of about 27.247 kilograms. Write how to say the nugget’s mass in words.

Answer:

the mass of gold nugget in Las Vegas is 27.247 kilograms given

In Words form it is twenty seven and two tenths  four hundredths and  seven thousandths

Review & Refresh

Compare.
Question 14.

Answer:

8/10 =0.8

80/100=0.8

both are equal 8/10 = 80/100

Question 15.

Answer:

5/8 = 0.625

3/6= 0.5

0.625 > 0.5,

5/8 > 3/ 6, 5/8 is greater than 3/6

Question 16.

Answer:

7/2= 3.5

10/8=1.25

so 7/2 is greater than 10/8.

7/2 > 10/8

Lesson 1.6 Compare Decimals

Explore and Grow

Use models to compare the decimals.

Answer:

0.62 >0.26, 0.62 is greater than 0.26, 0.6  is greater than o.2

0.80= 0.8, 0.80 both are  equal

3.5 < 3.55, 3.55 is greater than 3.50

Reasoning
How can you use a place value chart to compare two decimals? Use a place value chart to check your answers above.
Answer:

we use another table to compare with the previous and write the answer.

Think and Grow: Compare Decimals

Example
Compare 3.769 and 3.749.
Use a place value chart. Start at the left. Compare the digits in each place until the digits differ.

The digits in the ones place and the tenths place are the same. Compare the hundredths.

Example
Compare 2.4 and 2.405.
Use place value. Line up the decimal points. Start at the left. Compare the digits in each place until the digits differ.

2.4<2.405

Show and Grow

Compare
Question 1.

Answer:

So 9.063 < 9.067

Question 2.

Answer:

So 0.89 > 0.809,

Apply and Grow: Practice
Compare.
Question 3.

Answer:

8.537 < 8.541

Question 4.

Answer:

6.401 < 6.409, since  0.009 is greater than 0.001

Question 5.

Answer:

7.409 > 7.049 since 7.4 is greater than 7.0

Question 6.

Answer:

0.25 = 0.250

Both are equal

Question 7.

Answer:

2.701 >2.700, since 0.001 is greater than 0.000

Question 8.

Answer:

4.006 < 4.61, since 4.6 is greater than 4.0

Question 9.

Answer:

0.041 < 41.6, 41 is greater than 0

Question 10.

Answer:

0.007 < 0.7 as 0.7 is greater than 0.007

Order the decimals from least to greatest.
Question 11.
321.499, 325.499, 321.489
Answer:

321.499, 325.499, 321.489 from least to greatest

as 321.489 is smaller than 321.499 and 321.499 is smaller than 325.499

so 321.489 , 321.499 , 325.499

Question 12.
9.7, 9.64, 9.78
Answer:

9.7, 9.64, 9.78 from least to greatest

9.64 is smaller than 9.7 and  9.7 is smaller than 9.78

so 9.64, 9.7 , 9.78

Open-Ended
Complete the number to make the statement true.
Question 13.

Answer:

10.321 > 10.311

as 10.311 is smaller than 10.321

Question 16.

Answer:

28.60 = 28.600

Both are equal

Question 15.
Number Sense
Is 0.472 greater than or less than \(\frac{47}{1,000}\)? Explain.
Answer:

0.472, 0.047

0.475 is greater than 0.047,

as 4 in the tenths  place is greater than 0 in the others tenth place

Question 16.
YOU BE THE TEACHER
Your friend says that 45.6 is less than 45.57 because 6 is less than 57. Is your friend correct? Explain.
Answer:

Friend says 45.6 is less than 45.57

No ,he is wrong as the 6th in tenths place is greater than 5, in the tenths place

so he is wrong  45.60 > 45.57 not less

Think and Grow: Modeling Real Life

Example
You, your friend, and your cousin compete at a gymnastics competition. Your floor routine score is 15.633. Your friend’s score is 15.533, and your cousin’s score is 15.635. Order the scores from least to greatest.

Use a place value chart. Compare the digits in each place until the digits differ.

_15.533______ is the least
Write the remaining numbers in the place value chart. Compare the digits in each place until the digits differ.

15.633 ,15.635

as 15.533 is less than 15.633 as  5 at tenths place is less than 6 at tenths place

and 15.633 is less than 15.653 as 3 at hundredths place is less than 5 at hundredths place

So, the scores from least to greatest are 15.533, 15.633 and 15.635

Show and Grow

Question 17.
You stand on one leg for 2.75 minutes, your friend stands on one leg for 2 minutes, and your cousin stands on one leg for 2.25 minutes. Order the amounts of time from least to greatest.
Answer:

You – 2.75 min , Friend – 2.00 min and cousin for 2.25 min

2.00< 2.25 ,2.00 is less than 2.25 as 0 at tenths place is less than 2 at tenths place

2.25 < 2.75, 2.25 is less than 2.75 as 2 at tenths place is less than 7 at tenths place

so From Least to Greatest : 2.00 min, 2.25 min, 2.75 min

Question 18.
DIG DEEPER!
You, Newton, Descartes, and your friend each have a tablet. The table shows the screen display sizes. Your friend’s tablet has the greatest display size. Your tablet’s display size is greater than Newton’s but less than Descartes’s. What is the display size of your tablet.

Answer:

Y-Your, N-Newton’s ,F-Friend- 12.9, D-Descartes

Given Friend has greatest display size F= 12.9

Y – N ,given Your tablet’s display size is greater than Newton’s, N< Y and yours is less than Descartes’s Y < D

therefore  N < Y <D < F, Newton’s< Yours<Descartes’s<Friend

so N-Newton’s- 7.9, Y-Your-9.7,D-Descartes-10.5,F-Friend-12.9

Compare Decimals Homework & Practice 1.6

Write which place to use when comparing the numbers.
Question 1.
0.521
0.576
Answer:

2 at hundredths place  is smaller than 7 at hundredths place

so 0.521 < 0.576

Question 2.
17.422
17.946
Answer:

4 at tenths place  is small than 9 at tenths place

so 17.422 < 17.946

Question 3.
9.678
9.67
Answer:

8 at thousandths place is greater than 0 at thousandths place

9.678 > 9.670

Compare.
Question 4.

Answer:

4 at hundredths  place is smaller than 7 at hundredths place

so 3.445 is smaller than 3.472

3.445 < 3.472

Question 5.

Answer:

0 at tenths place is smaller than 4 at tenths place

so 23.049 < 23.409

Question 6.

Answer:

4 at tenths place is greater than 3 at tenths place

75.4 > 75.391

Question 7.

Answer:

All  given place values are same

so 14.100 or 14.10 = 14.100

Question 8.

Answer:

the value of 5s at hundredths place is more than 0 in other hundredths place

4.05> 4.005

Question 9.

Answer:

15.2, 15.002

2 at tenths place is greater than 0 at tenths place

15.2 > 15.002

Question 10.

Answer:

0.021, 0.026

1 at thousandths is smaller than 6 at thousandths place

0.021 < 0.026

Order the decimals from least to greatest.
Question 11.
2.75, 0.2, 0.275
Answer:

0.2 < 0.275 as 0 at hundredths place is less than 7 at hundredths place

0.275<2.75 as 0 at ones place is less than 2 at at ones place

so from least to greatest 0.2, 0.275, 2.75

Question 12.
56.01, 56.1, 56.001
Answer:

56.001 < 56.01 as 0 at hundredths place is less than 1 at hundredths place

56.01 < 56.1 as o at tenths place is less than 1 at tenths place

56.001 , 56.01, 56.1

Open-Ended
Complete the number to make the statement true.
Question 13.

Answer:

29.030 = 29.030

both the place values are same

Question 14.

Answer:

3.562 <3.562

as the value of  6 is at hundredths place so the other value at hundredths place is 6.

Question 15.
YOU BE THE TEACHER
Newton says 8.51 is less than 8.492 because 8.51 has fewer digits after the decimal point than 8.492. Is he correct? Explain.
Answer:

No, he is not correct its not the digits after the decimal point is fewer

but the 5s at tenths place is greater than 4 at the tenths place so

8.51 is greater than 8.492, 8.51 > 8.492

Question 16.
Open-Ended
Descartes is thinking of a number less than 46.922 and greater than 46.915. What could Descartes’s number be?
Answer:

The numbers can be 46.916,  4 6.917, 46.918, 46.919, 46.920 or 46.921 and

all these numbers are less than 46.922 and greater than 46.915.

Question 17.
Modeling Real Life
Player A’s batting average is 0.300, Player B’s batting average is 0.333, and Player C’s batting average is 0.313. Order the batting averages from greatest to least.
Answer:

B- 0.333, C-0.313 , A – 0.300

0.333 is great than 0.313 as 3 at hundredths place is great than 1 at hundredths place

so 0.333>0.313

0.313 is greater than 0.300 as 1 at hundredths place is great than 0 at hundredths place

0.313 > 0.300

the batting averages from greatest to least are B> C> A=0.333 > 0.313 >0.300

batting averages from greatest to least 0.333,0.313,0.300

Question 18.
Modeling Real Life
A gasoline station customer pumps more than 9.487 gallons of gasoline but less than 10 gallons. Which display could be his?

Answer:

Given A gasoline station customer pumps more than 9.487 gallons but less than 10 gallons

as 9.003 is less than 9.487 so not 9.003

as 9.499 is greater than 9.487  and even less than 10.000 so it is 9.499

as 9.406 is less than 9.487 so it cannot be 9.406

as 9.872 is greater than 9.487  and even less than 10.000 so it can be 9.872

so the displays can be  9.499 or 9.82

as both are more than 9.487 gallons and less than 10 gallons

Review & Refresh

Round the number to the place of the underlined digit.
Question 19.
7,851
Answer:

the round place of 5 is 7,850

Question 20.
9,462
Answer:

the round place of 9 means 10,000 or 9,500

Question 21.
4,983
Answer:

the round place of 9 – 5,000

Question 22.
51,504
Answer:

the round place of 1- 52,504

Lesson 1.7 Round Decimals

Explore and Grow

Plot the numbers on the number line. Which numbers round to 3? Which numbers round to4? How do you know?

Answer:

3,  3.09, 3.5, 3.51, 3.6, 3.77, 3.9 , 4

The numbers round to 3 are 3.09

The numbers round to 4 are 3.51, 3.6, 3.77 , 3.9

Repeated Reasoning
Show how you can use a number line to round 3.09, 3.51, and 3.77 to the nearest tenth.
Answer:

3.09 to the nearest tenth is 3.10

3.51 to the nearest tenth is 3.50

3.77 to the nearest tenth is 3.80

Think and Grow: Round Decimal Number

Example
Use a number line to round 7.36 to the nearest tenth.

7.36 is closer to 7.4 than it is to 7.3.
So, 7.36 rounded to the nearest tenth is __7.40____.

Example
Use place value to round 2.185 to the nearest whole number and to the nearest hundredth.

So, 2.185 rounded to the nearest whole number is __2.2_____.
2.185 rounded to the nearest hundredth is ___2.200___.

Show and Grow

Round the number to the place of the underlined digit.
Question 1.
12.67
Answer:

the round place of  digit 6 is 12.70

Question 2.
0.439
Answer:

the round place of  digit 4 is 0.5

the round place of  digit 3 is 0.44

the round place of digit  9 is 0.440

Question 3.
2.555
Answer:

the round place of  digit 2 is 3.000

Question 4.
5.409

the round place of digit 4 is 5.400

Question 5.
Round 0.68 to the nearest tenth.
Answer:

0.68 to the nearest tenth  is 0.70, 6 at tenths place becomes 7

Question 6.
Round 1.715 to the nearest hundredth.
Answer:

1.715 to the nearest hundredth is  1.720, 1 at hundredths place becomes 2

Question 7.
Round 4.07 to the nearest whole number.
Answer:

4.07 to the nearest whole number becomes  4.00 or 4

Question 8.
Round 0.289 to the nearest tenth.
Answer:

0.289 to the nearest tenth is 0.300 as 2 becomes 3 at tenths place.

Apply and Grow: Practice

Round the number to the place of the underlined digit.
Question 9.
1.482
Answer:

the underlined digit is 8 ,so its value becomes 1.490

Question 10.
5.093
Answer:

the underlined digit is 0 so its value  becomes 5.100

Question 11.
8.502
Answer:

the underlined digit is 8 so its value becomes 9.000

Question 12.
34.748
Answer:

if it is underlined at 3 it becomes 35.000

if it is underlined at 4 it becomes 35.000

if it is underlined at 7 it becomes 35.000

if it is underlined at 4 it becomes 34.800

if it is underlined at 8 it becomes 34.750

Question 13.
Round 2.619 to the nearest whole number.
Answer:

the value of 2.619 becomes 3.000

Question 14.
Round 7.825 to the nearest tenth.
Answer:

the value of 7.825 to the nearest tenth is 7.900

Question 15.
Round 92.701 to the nearest ten.
Answer:

the value of 92.701 to the nearest ten 93.000

Question 16.
Round 4.263 to the nearest hundredth.
Answer:

the value of 4.263 to the nearest hundredth is 4.270

Question 17.
Round 0.829.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Answer:

Round 0.829
Nearest whole number:0.900
Nearest tenth:0.830
Nearest hundredth:0.830

Question 18.
Round 18.062.
Nearest whole number:18.100
Nearest tenth:18.070
Nearest hundredth:18.063
Answer:

Question 19.
A baby harp seal weighs 25.482 pounds. Round this weight to the nearest tenth of a pound.

Answer:

Given a baby harp seal weighs 25.482 pounds and the nearest tenth is 25.000 pounds

So the weight of baby harp seal is 25.000 pounds.

Name the place value to which the number was rounded.
Question 20.
8.942 to 8.94
Answer:

It was rounded at hundredths value place

Question 21.
0.164 to 0.2
Answer:

It was rounded at tenths value place

Question 22.
15.826 to 16
Answer:

It was rounded at whole value place

Question 23.
Writing
Explain what happens when you round 2.999 to the nearest tenth.
Answer:

2.999 round value becomes 3.000 as all value places at tenths, hundredths, thousandths are 9 it becomes increased as we move so it ones value increases by 1 and becomes round 3.000.

Question 24.
DIG DEEPER!
To what place should you round 23.459 to get the greatest number? the least number? Explain.
Answer:

To make 23.459 to  greatest number the value of 4s at tenths  becomes 5,

23.500 and to make 23.500 round make 5 at tenths value increased and make ones value 3 as 4 so we get 24.000

and to make 23.459 to least number the value at hundredths 5 becomes 0 ,

23.400  and to make 23.400 round make 4 at tenths value as decreased to 0 and ones value same as 3 it becomes as 23.000

Think and Grow: Modeling Real Life

Example
Gasoline prices are listed to the nearest thousandth of a dollar. The final price is rounded to the nearest hundredth. About how much does a customer pay for 1 gallon of regular gasoline at the station shown?

Think: What do you know? What do you need to find? How will you solve?
Use place value to round the price of 1 gallon of regular gasoline, $2.799, to the nearest hundredth.

So, a customer will pay about $ ___$ 2.800____ for 1 gallon of gasoline.

Show and Grow

Use the table.

Question 25.
What is the length of the praying mantis rounded to the nearest hundredth?
Answer:

Given  praying mantis as 3.254 so the  nearest hundredth becomes 3.26

Praying mantis becomes 3.260

Question 26.
What is the length of the cicada rounded to the nearest tenth?
Answer:

Given cicada 1.48 so rounded to the nearest tenth becomes 1.50

Cicada becomes 1.50

Question 27.
What is the length of the hissing cockroach rounded to the nearest tenth?
Answer:

Given Hissing cockroach 2.682 so rounded to the nearest tenth is 2.700

Hissing cockroach becomes 2.700

Question 28.
DIG DEEPER!
You have about $3 in coins. Write one possible combination of coins that represents the least amount of money you could have. Write another combination of coins for the greatest amount of money you could have.
Answer:

$3 Least amount of money combinations – 1. $ 1.0, $1.0    2.$ 1.0, $1.5

3.$0.5, $ 2.0 all combinations becomes less than $3

$ 3 greatest amount of money combinations- 1. $1.0, $ 2.5   2.$1.5 ,$2.0

3. $2.0 , $ 2.0 all combinations becomes more than $3

Round Decimals Homework & Practice 1.7

Round the number to the place of the underlined digit.
Question 1.
49.012
Answer:

the underline digit is 4 its round number 4 becomes 5 so it is 50.000

Question 2.
2.308
Answer:

the underline digit is 2  its round number 2 becomes as 2.000

Question 3.
9.647
Answer:

the underline digit is 6 its round number 6 becomes 7 so it is 9.700

Question 4.
7.519
Answer:

the underline digit is 1 its round number 1 becomes  2 so it is 7.520

Question 5.
Round 8.436 to the nearest hundredth.
Answer:

8.436 to the nearest hundredth, 3 becomes 4 so it is 8.440

Question 6.
Round 15.159 to the nearest ten.
Answer:

15.159 to the nearest ten ,1 becomes 2 so it is 15.200

Question 7.
Round 1.602 to the nearest whole number.
Answer:

1.602 to the nearest whole number is 2.0

Question 8.
Round 3.619 to the nearest tenth.
Answer:

3.619 to the nearest tenth, so 6 becomes 7 it is 3.700

Question 9.
Round 4.183.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Answer:

Round 4.183.
Nearest whole number:4.000
Nearest tenth:4.200
Nearest hundredth:4.200

Question 10.
Round 9.076.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Answer:

Round 9.076.
Nearest whole number:9.000
Nearest tenth:9.100
Nearest hundredth:9.080

Name the place value to which each number was rounded.
Question 11.
16.932 to 20
Answer:

16.932 at Tens value it is rounded  so it becomes 20

Question 12.
0.581 to 0.58
Answer:

0.581 to 0.58

0.581 at Thousandths value has been rounded so 0.58

Question 13.
7.429 to 7.4
Answer:

7.429 to 7.4

7.429 at Hundredths value has been rounded so 7.4

Question 14.
Structure
Round ★ to the nearest tenth.

Answer:

the nearest tenth * in the number line is showing at 5.64

Question 15.
Precision
The area of a campground is exactly halfway between 25.9 acres and 26 acres. What is the area of the campground?
Answer:

Halfway of 25.9 and 26.0 is 25.90+25.60=51.50/2 = 25.75 acres

So the area of the campground is 25.75 acres.

Question 16.
Open-Ended
Name two different numbers that round to 3.8 when rounded to the nearest tenth.
Answer:

The two numbers that are round to 3.8 when rounded to the nearest tenth place the value becomes 4.0 and  3.90

Question 17.
Open-Ended
Name two different numbers that round to 7.42 when rounded to the nearest hundredth.
Answer:

The two numbers that round to 7.42 when rounded to the nearest hundredth place the value becomes 7.50 and 7.40

Use the table.

Question 18.
Modeling Real Life
Your science class designs and tests four model boats to find out how much weight they can hold without sinking. What is the greatest weight rounded to the nearest tenth that a boat can hold?
Answer:

Given weights held without sinking in kilograms are

0.694,0.605,0.592,0.547 among all the weights the greatest weight rounded to the nearest tenth that a boat can hold is (0.694) i.e  0.7 kilograms

Question 19.
Modeling Real Life
What is the least weight rounded to the nearest hundredth that a boat can hold?
Answer:

The least weight rounded to the nearest hundredth is (0.547)- o.5 kilograms a boat can hold

Review & Refresh

Find the product.
Question 20.
7 × 40
Answer:

The product of 7 X 40 = 280

Question 21.
5,000 × 9
Answer:

The product of 5,000 x 9 = 45,000

Question 22.
8 × 200
Answer:

The product of 8 x 200=1,600

Place Value Concepts Performance Task

There are 18 species of penguins. Scientists have estimated the populations of 16 penguin species.
Question 1.
What fraction of penguin species have unknown populations?
Answer:

Given total is 18 species of penguins out of which Scientists have estimated the populations of 16 penguin species. so unknown is 18-16/18=2/18,

so 2/18=1/9 of penguin species have unknown populations

Question 2.
Several species of penguins and their estimated populations and locations are shown.

a. Are there more emperor penguins or rockhopper penguins? Explain.
b. Which species of penguin has the greatest population? Explain.
c. About how many penguins live in Antarctica? Round your answer to the nearest hundred thousand.

d. The Galápagos penguin is an endangered species. There are about 1,000 times as many macaroni penguins as Galápagos penguins. About how many Galápagos penguins are there?
Answer:

a. Rock hopper penguins are more than Emperor

2,460,000>595,000 as  2,460,000 is greater than 595,000 so Rock hopper penguins are more

B. Macaroni species 18 x 10⁶ of penguin has the greatest population as compared to Emperor 595,000,Adelie 4,000,000+7,00,000+40,000=4,740,000 and

Rockhopper-2,460,000

among all the species Macaroni 18 x 10⁶– species is more.

C. Macaroni + Adelie + Rockhopper +Emperor

18,000,000+4,740,000+ 2,460,000 + 595,000 = 25,795,000

25,785,000  to the nearest hundred thousand is 26,000,000

So there are almost 26,000,000 penguins live in Antarctica.

d. 1,000 times as many macaroni penguins as Galápagos penguins is

18 x 10^{6  x} 1 x 1000 = 18 x 10⁹ 

So there are 18 x 10⁹ Galápagos penguins available

Place Value Concepts Activity

Place Value Plug In
Directions:
1. Players take turns.
2. On your turn, roll six dice. Arrange the dice into a six-digit number that matches one of the descriptions.
3. Write your number on the lines.
4. The first player to fill in all of the numbers wins!

Answer:

Place Value Concepts Chapter Practice

1.1 Place Value Patterns

Use a place value chart to answer the question.
Question 1.
What number is 10 times as great as 4,000?
Answer:

10 times as great as 4,000 is 10 x 4,000= 40,000

Question 2.
What number is \(\frac{1}{10}\) of 8,000?
Answer:

\(\frac{1}{10}\) of 8,000 is 8,000/10 = 800

Question 3.
10,000 is 10 times as great as what number?
Answer:

10,000 is 10 times as great as 1,000

Question 4.
70 is \(\frac{1}{10}\) of what number?
Answer:

70 is \(\frac{1}{10}\) of 700

Question 5.
Complete the table.

Answer:

300 is 10 times as great as 30  and 30 x 1/10 is 3

6,000 is 10 times as great as 600 and 600 x 1/10  is 60

90,000 is 10 times as great as 9,000 and 9,000 x 1/10 is 900

2,00,000 is 10 times as great as 20,000 and 20,000 x 1/10=2,000

Question 6.
YOU BE THE TEACHER
Your friend says 500 is 10 times as great as 5,000. Is your friend correct? Explain.
Answer:

No, My friend is wrong because  500 is not 10 times great as 5,000,

500 < 5,000.

1.2 Place Value with Whole Numbers

Question 7.
Write the number in two other forms.
Standard form: 456,701
Word form:
Expanded form:
Answer:

Word form: Four hundred fifty six thousand, seven hundred and one
Expanded form:4 x 100000 + 5 x 10000 + 6 x 1000 + 7 x 100 + 1

Question 8.
Write the number in two other forms.
Standard form:
Word form: Eight million, sixty thousand, five hundred seventy-three
Expanded form:
Answer:

Standard form:8,060,573
Expanded form:8 x 1000000+6 x 10000 + 5 x 100 + 7 x 10 + 3

Question 9.
Compare the values of the 4s in the number 900,441,358.
Answer:

4s value is at lakh or hundred thousandths place and another 4s place is at ten thousandths place.

Question 10.
Write the values of the 6s in the number 96,672.
Answer:

The values of the 6s in the number 96,672 are

6s place is at thousand and another 6s is at hundreds place

Compare.
Question 11.

Answer:

83,802 > 83,082

The value at hundred 8 is more/great to 0 at hundreds place

Question 12.

Answer:

2,498,576 > 2,477,583

The value at 9 at ten thousands place is more/great than 7 at ten thousands place

1.3 Patterns and Powers of 10

Question 13.
Write 10 × 10 as a power.
Answer:

10 × 10 as a power is 10²

Find each product. Use patterns to help.
Question 14.
4 × 10 = _____
4 × 100 = ______
4 × 1,000 = _____
4 × 10,000 = ______
Answer:

4 × 10 = 40
4 × 100 = 400
4 × 1,000 = 4,000
4 × 10,000 =40,000

Question 15.
3 × 10 = _____
3 × 100 = ______
3 × 1,000 = _____
3 × 10,000 = ______
Answer:

3 × 10 = 30
3 × 100 = 300
3 × 1,000 = 3,000
3 × 10,000 = 30,000

Question 16.
7 × 10 = _____
7 × 100 = ______
7 × 1,000 = _____
7 × 10,000 = ______
Answer:

7 × 10 = 70
7 × 100 = 700
7 × 1,000 = 7,000
7 × 10,000 = 70,000

Find the value of the expression.
Question 17.
10⁵
Answer:

10⁵=1,00,000

Question 18.
8 × 10¹
Answer:

8 × 10¹⁼ 8 x 10 = 80

Question 19.
7 × 10⁴
Answer:

7 × 10⁴ = 7 x  10 x 10 x 10 x 10= 70,000

Question 20.
3 × 10⁵
Answer:

3 × 10⁵ = 3 X 10 X 10 X 10 X 10 X 10=3,00,000

Rewrite the number as a whole number multiplied by a power of 10.
Question 21.
5,000
Answer:

5,000= 5 X 10³

Question 22.
600,000
Answer:

6,00,000= 6 X 10⁵

Question 23.
90
Answer:

90= 90 X 10¹

1.4 Decimals to Thousandths

Write the decimal as a fraction.
Question 24.
0.062
Answer:

0.062= 62 X 1/1,000

Question 25.
0.008
Answer:

0.008= 8 X 1/1,000

Question 26.
0.195
Answer:

0.195= 195 X 1/1,000

Write the fraction as a decimal.
Question 27.
\(\frac{2}{1,000}\)
Answer:

\(\frac{2}{1,000}\) = 2 x 1/1,000= 0.002

Question 28.
\(\frac{37}{1,000}\)
Answer:

\(\frac{37}{1,000}\) = 37 x 1/1,000= 0.037

Question 29.
\(\frac{409}{1,000}\)
Answer:

\(\frac{409}{1,000}\)= 409/1,000=0.409

Question 30.
0.0.7 is 10 times as great as what number?
Answer:

0.07 is 10 times great as 0.007

Question 31.
0.04 is \(\frac{1}{10}\) of what number?
Answer:

0.04 is \(\frac{1}{10}\) of 0.4

1.5 Place Value with Decimals

Write the number in two other forms.
Question 32.
Standard form:
Word form:
Expanded form: 5 × 1 + 3 × \(\frac{1}{10}\) + 8 × \(\frac{1}{100}\) + 4 × \(\frac{1}{1,000}\)
Answer:

Standard form:5+0.3+0.08+0.004=5.384
Word form: five and three tenths, eight hundredths and 4 thousandths

Question 33.
Standard form: 2.059
Word form:
Expanded form:
Answer:

Word form: two and five hundredths and nine thousandths

Expanded form: 2 x 1 + 5 x 1/100 + 9 x 1/1000

Question 34.
Compare the values of the 5s in the number 1.055.
Answer:

5s value is at hundredths and another 5s value is at thousandths place

Question 35.
Compare the values of the 8s in the number 6.884.
Answer:

8s place value is at tenths value and other 8s place value is at hundredths place

1.6 Compare Decimals

Compare.
Question 36.

Answer:

15.891 > 15.791

the 8 value at tenths place is greater than 7 at tenths place

Question 37.

Answer:

8.205 < 8.250

the 0 at hundredths is less than 5 at hundredths

Question 38.

Answer:

Both are same

3.600 = 3.6

Order the decimals from least to greatest.
Question 39.
7.008, 7.09, 7.180
Answer:

7.008<7.09

7.09 <7.18

so the decimals from least to greatest are 7.008 , 7.09, 7.180

Question 40.
50.426, 50.42, 50
Answer:

50 < 50.42

50.42 < 50.426

so the decimals from least to greatest are 50, 50.42, 5.0426

Question 41.
Modeling Real Life
Newton weighs a treat at the pet store. He says it weighs less than 0.519 ounce but more than 0.453 ounce. Which treats could he have weighed?

Answer:

Newton says it weighs less than 0.519 ounce  but more than 0.453 ounce,

so Blue berry waffle 0.512 which is in between 0.453<0.512<0.519

and We have more than 0.453 is 0.459 ounce which is Peanut butter and is less than 0.519 ounce

So Peanut butter 0.453 is in between 0.453<0.459<0.519

so newton would have weighed Peanut butter 0.459 ounce

and it can be even Blueberry  waffle 0.512  ounce

newton’s treats could be Peanut butter ,Blueberry waffle

1.7 Round Decimals

Round the number to the place of the underlined digit.
Question 42.
9.514
Answer:

if 9 is digit then it is rounded as 9.0

if 5 then  it is rounded as 9.500

if 1 then it is rounded as 9.520

if  4 then it is rounded as 9.520,9.510

Question 43.
1.027
Answer:

1.027 at 2 it is rounded as 1.03

Question 44.
8.469
Answer:

8.469, 8 is rounded as 8.500 or 8 or 8.000

Question 45.
32.501
Answer:

in 32.501

3 is rounded as  30.000

2- 33.000

5- 32.6 or 33

0-32.5

1 -32.500

Question 46.
Round 0.176 to the nearest
Answer:

0.200

Question 47.
Round 6.538 to the nearest tenth. hundredth.
Answer:

6.500 

Question 48.
Round 7.425.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Answer:

Nearest whole number:8.000, 8
Nearest tenth:7.500
Nearest hundredth:7.430

Question 49.
Round 2.108.
Nearest whole number:
Nearest tenth:
Nearest hundredth:
Answer:

Nearest whole number:2.000, 2
Nearest tenth:2.100
Nearest hundredth:2.100

Final Words:

It is very important for the students to understand and learn the fundamentals at the primary level itself. Here we have prepared the questions as per the latest edition 2019. Keep the textbook aside and try to solve the problems by referring to our Big Ideas Math Answers Grade 5 Chapter 1 Place Value Concepts. To make you comfortable we have provided the solution key for Big Ideas Math Grade 5 Chapter 1 Place Value Concepts in the pdf format. Stay tuned to our CCSS Math Answers to get the latest updates of BIM Grade 5 Chapters.

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Big Ideas Math Book 8th Grade Answer Key Chapter 7 Functions

Big Ideas Math Grade 8 Ch 7 Functions Solutions provided here cover the questions from Lessons 7.1 to 7.5, Practice Tests, Review Tests, Cumulative Practice, Assessment Tests, etc. You can have deeper insights into all the Topics of Functions such as the representation of functions, linear functions, analyzing and sketching graphs, etc. Use the below available quick links for Big Ideas Math 8th Grade Chapter 7 Functions Answer Key and clear all your ambiguities regarding the concerned topics.

STEAM Video/Performance Task

• STEAM Video/Performance Task

Getting Ready for Chapter 7

• Getting Ready for Chapter 7

Lesson 1 Relations and Functions

• Lesson 7.1 Relations and Functions
• Relations and Functions Homework & Practice 7.1

Lesson 2 Representations of Functions 

• Lesson 7.2 Representations of Functions
• Representations of Functions Homework & Practice 7.2

Lesson 3 Linear Functions

• Lesson 7.3 Linear Functions
• Linear Functions Homework & Practice 7.3 

Lesson 4 Comparing Linear and Non Linear Functions

• Lesson 7.4 Comparing Linear and Non Linear Functions
• Comparing Linear and Non Linear Functions Homework & Practice 7.4

Lesson 5 Analyzing and Sketching Graphs

• Lesson 7.5 Analyzing and Sketching Graphs
• Analyzing and Sketching Graphs Homework & Practice 7.5

Functions Connecting Concepts

• Functions Connecting Concepts
• Functions Chapter Review
• Functions Practice Test
• Functions Cumulative Practice

Functions STEAM Video/Performance Task

STEAM Video

Apparent Temperature
Sometimes it feels hotter or colder outside than the actual apparent temperature. How hot or cold it feels is called the temperature. What weather factors might contribute to the apparent temperature?
Watch the STEAM Video “Apparent Temperature.” Then answer the following questions.

1. Robert says that the Wet-Bulb Globe Temperature (WBGT)index is used as a measure of apparent temperature.

In the formula, T_W is the natural wet-bulb temperature, T_G is the black-globe temperature, T_D and is the dry-bulb temperature. Find WBGT when T_W = 75ºF, T_G = 100ºF, and T_D = 84ºF.
2. Different categories of Wet-Bulb Globe Temperatures are shown in the chart. Each category can be represented by a different-colored flag. Which flag color is displayed when WGBT = 87.5ºF?

Performance Task

Heat Index
After completing this chapter, you will be able to use the STEAM concepts you learned to answer the questions in the Video Performance Task. You will be given information about heat index.

You will be asked to create a graph of the temperatures and heat indices. Why is it useful to know the heat index?

Functions Getting Ready for Chapter 7

Chapter Exploration

Work with a partner. Copy and complete the diagram.

1.  Answer: ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Area A of a rectangle we have , A = w × l
for x = 1 , A = 2 × 1 = 2 ,
for x = 2 , A = 2 × 2 = 4 ,
for x = 3 , A = 2 × 3 = 6 ,
for x = 4 , A = 2 × 4 = 8 ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output A = 2 , 4 , 6 , 8 , respectively .
That is ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) .

2. Answer: ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ).

Explanation:
Given , Width w = 2 , Length l = x  where x = 1 , 2 , 3 , 4 .
To find Perimeter of a rectangle we have , P = 2( l + w )
for x = 1 ,P = 2( 1 + 2 ) = 2 × 3 = 6 ,
for x = 2 , P = 2( 2 + 2 ) = 2 × 4 = 8  ,
for x = 3 , P = 2( 3 + 2 ) = 2 × 5 = 10  ,
for x = 4 , P = 2( 4 + 2 ) = 2 × 6 = 12  ,
So, for every value of Input x = 1 , 2 , 3 , 4 we have Output P = 6 , 8 , 10 , 12 , respectively .
That is ( 1 , 6 ) , ( 2 , 8 ) , ( 3 , 10 ) , ( 4 , 12 ) .

3. Answer : ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

Explanation:
Given , Radius of a circle , where as  r = 1 , 2 , 3 , 4
To find the circumference of a circle , we have C = 2Òr , Ò = 3.14 , or we can write it as 3 .
for r = 1 , C = 2 × 3 × 1 = 6 ,
for r = 2 , C = 2 × 3 × 2 = 12 ,
for r = 3 , C = 2 × 3 × 3 = 18 ,
for r = 4 , C = 2 × 3 × 4 = 24 ,
So, for every value of Input r = 1 , 2 , 3 , 4 we have Output C = 6 , 12 , 18 , 24 , respectively .
That is ( 1 , 6 ) , ( 2 , 12 ) , ( 3 , 18 ) , ( 4 , 24 ) .

4. Answer: ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 )

Explanation:
Given , Two Edges of a cube = 3 , h = 1 , 2 , 3 , 4
To find the Volume of the cube we have , V = a³
for h = 1 , V = 3 × 3 × 1 = 9 ,
for h = 2 , V = 3 × 3 × 2 = 12 ,
for h = 3 , V = 3 × 3 × 3 = 27 ,
for h = 4 , V = 3 × 3 × 4 = 36 ,
So, for every value of Input h = 1 , 2 , 3 , 4 we have Output V = 9 , 18 , 27 , 36 , respectively .
That is ( 1 , 9 ) , ( 2 , 18 ) , ( 3 , 27 ) , ( 4 , 36 ) .

Vocabulary
The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.
input
mapping diagram
nonlinear function
output
linear function

Answer : Input : The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

mapping diagram : A relation pairs inputs with outputs , A relation can be represented by ordered pairs or a mapping diagram .

nonlinear function : nonlinear functions are functions which are not linear. Quadratic functions are one type of nonlinear function. It is a relation between two variables , function that does not form a line when graphed.

output ; The ordered pairs can be used to show inputs and outputs of a function ,The input is the number you feed into the expression, and the output is what you get after the calculations are finished.

linear function : A linear function is a relation between two variables that produces a straight line when graphed. And it has one dependent variable and one independent variable .

Lesson 7.1 Relations and Functions

EXPLORATION 1

Interpreting Diagrams
Work with a partner. Describe the relationship between the inputs and outputs in each diagram. Then complete each diagram. Is there more than one possible answer? Explain your reasoning.

Answer: a. The relation between the inputs and outputs is outputs are the result of twice as many times the inputs.
b. The relation between the inputs and outputs is outputs are the result of colors of inputs . In this case we can notice that , for any one input we can have more than one output .

Explanation:
a. As shown in the diagrams , The relation between the inputs and outputs is outputs are the result of twice as many times the inputs , so for input 1 = 1 × 1 = 1 as output ,
for input 2 = 2 × 2 = 4  ,
for input 3 = 3 × 3 = 9  ,
for input 5 = 5 × 5 = 25 ,
for input 8 = 8 × 8 = 64 ,
for input 9 = 9 × 9 = 81 ,

So, for every value of Input = 1 , 2 , 3 , 5 , 8 , 9 , we have Output = 1 , 4 , 9 , 25 , 64 , 81 , respectively .
That is ( 1 , 1 ) , ( 2 , 4 ) , ( 3 , 9 ) , ( 5 , 25 ) , ( 8 , 64 ) , ( 9 , 81 ) .

b. The relation between the inputs and outputs is outputs are the result of colors of inputs . 
for input Blueberry = color is blue as output
for  input lemon = color is yellow as output
for input Apple = color is yellow , red and green as output
for input Grape = color is green as output.

In this case we can notice that , for any one input we can have more than one output .

EXPLORATION 2

Describing Relationships Between Quantities
Work with a partner. The diagrams show the numbers of tickets bought by customers for two different plays and the total costs (in dollars).

a. For each diagram, how many outputs does each input have?
b. Describe the prices of tickets for each play.
c. A person buys 4 tickets for each play. Can you determine the total cost of all 8 tickets? Explain.
Answer:
a. For Play A ,The number of inputs are equal to number of outputs ,
For Play B , The number of inputs are not equal to number of outputs .
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A , The price of the each ticket is $8 .
For Play B , The price of each ticket is $4 or $8 .

c. For Play A , each ticket is $8 , Then for 4 tickets = 4 × $8 = $32 .
For Play B , each ticket is $4 or $8 , Then for 4 tickets = 4 × $8 = $32 . or 4 × $4 = $16 .

Explanation:
a. For Play A ,
The number of inputs are equal to number of outputs , 4 inputs = 4 outputs
That is ( 1 , 8 ) , ( 2 , 16 ) , ( 3 , 24 ) , ( 4 , 32 ) .
For Play B ,
The number of inputs are not equal to number of outputs , 3 inputs are not equal to 7 outputs
That is , for input 1 = 4 , 8 as outputs ,
for input 2 = 8 , 12 , 16 as outputs ,
for input 3 = 12 , 16 , 20 , 24 as outputs ,
So, input 1 = 2 outputs , input 2 = 3 outputs , input 3 = 4 outputs .

b. For Play A ,
The price of the each ticket is $8 .
For Play B ,
The price of each ticket is $4 or $8 .

c. Given , A person buys 4 tickets for each play.
For Play A , each ticket is $8 , Then for 4 tickets = 4 × $8 = $32 .
And for 8 tickets = 8 × $8 = $64 .
For Play B , each ticket is $4 or $8 , Then for 4 tickets = 4 × $8 = $32 . or 4 × $4 = $16 .
And for 8 tickets =8 × $4 =$32  or 8 × $8 = $64 .

Try It

List the ordered pairs shown in the mapping diagram.
Question 1.

Answer: Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 12 ) , ( 2 , 10 ) , ( 4 , 8 ) , ( 6 , 6 ) .

Question 2.

Answer: Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , -1 ) , ( 1 , -2 ) , ( 2 , -3 ) , ( 2 , -4 ) .

Determine whether the relation is a function.
Question 3.

Answer: The relation is not a function

Explanation:
The each input has  more than two outputs , Even one of those inputs are unclear of outputs
So , The relation is not a function .

Question 4.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
PRECISION
Describe how relations and functions are different.
Answer: Relations are nothing but the ordered pairs with Inputs and Outputs . On the other hand , Functions are The relation that pairs with one input with exactly one output  are called Functions.

IDENTIFYING FUNCTIONS List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 6.

Answer: The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 10 , 1 ) , ( 15 , 1 ) , ( 20 , 13 ) , ( 25 , 7 ) .
Each input has exactly one output ,
So, The relation is a function .

Question 7.

Answer: The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) and relation is not a function .

Explanation:
As shown , The ordered pairs are ( 0 , -5 ) , ( 0 , -4 ) , ( 1 , -4 ) , ( 2 , -3 ) , ( 3 , -2 ) .
The input 0 has more than one output ,
So, The relation is not a function .

Question 8.
OPEN-ENDED
Copy and complete the mapping diagram at the left to represent a relation that is a function. Then describe how you can not modify the mapping diagram so that the relation is a function.

Answer: ordered pairs are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) . To have the relation as a function we must have only one output for one input.

Explanation:
The ordered pairs of the diagram are ( -8 , -4 ) , ( 0 , -2 ) , ( 8 , 0 ) , ( 16 , 2 ) .
Each Input must have only one output in order to be the relation is a function ,
If ,The mapping diagram has the right to left representation or each input has more than one output , then the relation is not a function .
So , To have the relation as a function we must have only one output for one input.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 9.
The mapping diagram represents the costs of reserving a hotel room for different numbers of nights.

a. Is the cost a function of the number of nights reserved?
b. Describe the relationship between the cost and the number of nights reserved.
Answer: a. Yes , The cost is a function of the number of nights reserved .
b. The relationship between the cost and the number of nights reserved is , For every night reservation of the room is increasing by $85 with increase in the next reservation ,

Explanation:
a. From the diagram we have ,
Ordered pairs are ( 1 , -$85 ) , ( 2 , $170 ) , ( 3 , $255 ) , ( 4 , $340 ) . each input has exactly one output ,
So , the relation is a function and ,
Yes , The cost is a function of the number of nights reserved .

b. The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation,
that is , input 1 = $85 as output
Input 2 = $85 + $85 = $170 as output
Input 3 = $170 + $85 = $255 as output
Input 2 = $255 + $85 = $340 as output

So, The relationship between the cost and the number of nights reserved is ,
For every night reservation of the room is increasing by $85 with increase in the next reservation.

Question 10.
DIG DEEPER!
The graph represents the number of contestants in each round of a talent competition.
a. Is the number of contestants a function of the round number?
b. Predict the number of contestants in the talent competition during Round 7. Explain your reasoning.

Answer: a. The number of contestants is a function of the round number.
b. The number of contestants in the talent competition during Round 7  are 2.

Explanation:
a. From the given graph , The ordered pairs are ( 1 , 128 ) , ( 2 , 64 ) , ( 3 , 32 ) , ( 4 , 16 ) .
Each input has only one output , The relation is a function .
So , the number of contestants is a function of the round number.

b. Firstly , The relation between the input and output is,
With every increase in the round number the number of contestants are decreasing by half the number of the previous round , That is, for  input 1 = 128 as output
For input 2 = 128 – 64 = 64  as output
For input 3 = 64 – 32 = 32  as output
For input 4 = 32 – 16 = 16  as output
For input 5 = 16 – 8 = 8  as output
For input 6 = 8 – 4 = 4  as output
For input 7 = 4 – 2 = 2  as output,
So, The number of contestants in the talent competition during Round 7  are 2 .

Relations and Functions Homework & Practice 7.1

Review & Refresh

Choose an appropriate data display for the situation. Explain your reasoning.
Question 1.
the number of runners in each
age group at a marathon
Answer: In a marathon ,the people of all age group are participating for a promotion on healthy lifestyle, The number of runners in each  group has kids, adults and old people to spread the awareness of leading a healthy life by running daily in the morning . Running or jogging in the morning can help us to maintain our body mass index at an optimal level which is good for heart. The Marathon is conducted by the government of health ministry to be example for the future generations.

Question 2.
the high temperature and the
attendance at a water park each day
Answer:  Generally, The water park is normally crowded depending on the season and the temperature, In summer the attendance in the waterpark is at the utmost point because of the high temperature and the seasonal vacation. Going to the water park in summer is super fun due to the number of  water slides , water rides will be a nice place to the whole family trip and as well as friends . In order to be there at a less crowded time spring is also a nice time to visit the water park .

Graph the linear equation.
Question 3.
y = 2x – 3
Answer:
Explanation:
Given , y = 2x – 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 1 , then y = 2(1) – 3 = 2 – 3 = -1 . co-ordinates are (1 , -1)
if x = 2 , then y = 2(2) – 3 = 4 – 3 = 1 , co-ordinates are (2 , 1)
The co-ordinates (1 , -1) , (2 , 1) form a straight line .
So, y = 2x – 3 is a linear equation.

Question 4.
y = – 0.5x
Answer:
Explanation:
Given , y = -0.5x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) = 0 . co-ordinates are (0 , 0)
if x = 2 , then y = -0.5(2) = -1 , co-ordinates are (2 , -1)
The co-ordinates (0 , 0) , (2 , -1) form a straight line .
So, y = -0.5x is a linear equation.

Question 5.
y = – 3x + 4
Answer:
Explanation:
Given , y = – 3x + 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 3(0) + 4 = 4 . co-ordinates are (0 , 4)
if x = 1 , then y = – 3(1) + 4 = -3 + 4 = 1 , co-ordinates are (1 , 1)
if x = 2 , then y = – 3(2) + 4 = -6 + 4 = -2 , co-ordinates are (2 , -2)
The co-ordinates (0 , 4) , (1 , 1) , (2 , -2) form a straight line .
So, y = – 3x + 4 is a linear equation.

Question 6.
Which word best describes two figures that have the same size and the same shape?
A. congruent
B. adjacent
C. parallel
D. similar
Answer:  A. congruent

Explanation:
Two figures which have the same size and shape are congruent.

Concepts, Skills, &Problem Solving

INTERPRETING DIAGRAMS Describe the relationship between the inputs and outputs in the diagram. Then complete the diagram. Is there more than one possible answer? Explain your reasoning. (See Exploration 1, p. 275.)
Question 7.

Answer: The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output.

Explanation:
The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output ,
for input 1 = -1 as output
for input 2 = -1 + (-4) = -5 as output
for input 3 = -5 + (-4) = -9 as output
for input 4 = -9 + (-4) = -13 as output
for input 5 = -13 + (-4) = -17 as output
for input 6 = -17 + (-4) = -21 as output.
So, The relationship between the inputs and outputs in the diagram is ,
For every increase in number of input is having the output of adding -4 to the previous output .

In this case , we are witnessing only one output for one input.

Question 8.

Answer: The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.

Explanation:
The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.
For input basketball = b as output
For input baseball = b as output
For input football = f as output
For input soccer = s as output
For input swimming = s as output,
So, The relationship between the inputs and outputs in the diagram is,
Each input has the sports name and the output has the Starting letter of sports name.

In this case we have more than one output for input.

LISTING ORDERED PAIRS List the ordered pairs shown in the mapping diagram.
Question 9.

Answer: Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) .

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 0 , 4 ) , ( 3 , 5 ) , ( 6 , 6 ) , ( 9 , 7 ) .

Question 10.

Answer: Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ).

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 1 , 8 ) , ( 3 , 8 ) , ( 3 , 4 ) , ( 5 , 6 ) , ( 7 , 2 ).

Question 11.

Answer: Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ).

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 6 , -5 ) , ( 7 , -5 ) , ( 8 , -10 ) , ( 9 , -10 ).

IDENTIFYING FUNCTIONS Determine whether the relation is a function.
Question 12.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 0 has two outputs which are 10 and 20 .
So , The relation is not a function .

Question 13.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 14.

Answer: The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 15.
YOU BE THE TEACHER
Your friend determines whether the relation shown in the mapping diagram is a function. Is your friend correct? Explain your reasoning.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 4 has four outputs which are 5, 6 , 7 and 8.
So , The relation is not a function .

REASONING Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.
Question 16.

Answer:  The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( 1 , 1 ), ( 3 , 3 ), ( -1 , -1 ), ( -3 , -3 ).
Each input has exactly one output ,
So , The relation is a function .

Question 17.

Answer: The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( 0 , 8 ),( 2 , 8 ),( 4 , 8 ),( 6 , 8 ),( 8 , 8 ),( -2 , 8 ),( -4 , 8 ). Each input has exactly one output ,
So , The relation is a function.

Question 18.

Answer: The mapping diagram representing the relation is

Explanation:
From the given graph , co-ordinates of the ordering pairs are( -2 , 1 ),( -2 , 2 ),( -2 , 3 ),( -2 , 4 ),( -2 , 5 ),( -2 , 6 ).
Each input has more than one output ,
So , The relation is not a function.

Question 19.
MODELING REAL LIFE
The normal pressure at sea level is 1 atmosphere of pressure(1 ATM). As you dive below sea level, the pressure changes. The mapping diagram represents the pressures at different depths.

a. Complete the mapping diagram.
b. Is pressure a function of depth?
c. Describe the relationship between pressure and depth.
d. List the ordered pairs. Then plot the ordered pairs in a coordinate plane. What do you notice about the points?
e. RESEARCH What are common depths for beginner scuba divers? What are common depths for experienced scuba divers?
Answer: The detailed explanation of  all the answers are given below .

Explanation:
a. The mapping diagram is
b. Yes , the pressure is a function of depth, Because depth is related to pressure in the given mapping diagram.

c. The relationship between pressure and depth is,
for every 10m increase in Depth of input there is an increase in 1 ATM pressure .

d. The ordered pairs are ( 0 , 1 ) , ( 10 , 2 ) , (20 , 3 ) , ( 30 , 4 ) , ( 40 , 5 ), ( 50 , 6 ).
The plot of the ordered pairs in a coordinate plane is

From the graph, we have seen that, if the depth of the diving of scuba drivers increases then the water pressure increases with increase in depth. So, the graph have straight line .

e. The common depths for beginner scuba divers is 30 feet to 60 feet or 9 to 18 meters ,
The common depths for experienced scuba divers is more than 60 feet or more than 18 meters .

Question 20.
DIG DEEPER!
The table shows the cost of purchasing 1, 2, 3, or 4 T-shirts from a souvenir shop.

a. Is the cost a function of the number of T-shirts purchased?
b. Describe the relationship between the cost and the number cost per T-shirt of T-shirts purchased. How does the change as you purchase more T-shirts?
Answer: The detailed explanation of  all the answers are given below .

Explanation:
a. Yes , The cost is a function of the number of T-shirts purchased, Because the cost of the purchased T-shirts is varying with the number of T-shirts purchased.

b. The relationship between the cost and the number cost per T-shirt of T-shirts purchased is,
Input is the cost of 1 T-shirt is $10 as output , Then for 2 T-shirts cost will be $20
If 2 T-shirts will be purchased at same time, cost will be decreased by $2 so it will be $10 + 8 = $18 for 2 T-shirts.
As per the single T-shirt cost , For 3 T-shirts will be $30,
So in the table given that 3 T-shirts will cost $24 , because it cost $18 + 6 = $24 for 3 T-shirts.
It goes same for 4 T-shirts , For 4 T-shirts will be $40, because it cost $24 + 4 = $28 for 4 T-shirts.

The change as you purchase more T-shirts is For every increase in purchase of the number of T-shirts is decrease in the cost of total T-shirts purchased.

Question 21.
REPEATED REASONING
The table shows the outputs for several inputs. Use two methods to predict the output for an input of 200.

Answer: The output for an input of 200 is 1025.

Explanation:
Method 1. The relation between inputs and outputs is as follows,
y = 25 + 5x
As input increases by 1 , output increases by 5 units,
To find output of 200 as input ,
put x = 200 in the equation,
y = 25 + 5(200)
= 25 + 1000
= 1025.
So , y = 1025.

Method 2.  As the table shown, for every increase in input there is an increase in 5 numbers in output,
So , For 1 input = 25 + 5 = 30 as output
For 2 input = 30 + 5 = 35 as out put
For 3 input = 35 + 5 = 40 as out put
For 4 input = 40 + 5 = 45 as out put
By doing this for number 200 as input we have , 1025 as output.

Lesson 7.2 Representations of Functions

EXPLORATION 1

Using a Table to Describe Relationships
Work with a partner. Make a table that shows the relationship  between the figure number x and the area A of each figure. Then use an equation to find which figure has an area of 81 square units when the pattern continues.

Answer: a. The equation is y = 2x – 1, For figure has an area of 81 square units is 41.
b. The equation is y = x²,  For figure has an area of 81 square units is 9.

Explanation:
a. figure shows the 1 square unit of each box  for and it has a pattern of  2x – 1
figure 1 = 1 square unit
figure 2 =3 square units
figure 3 = 5 square unit and so on
So, the equation is y = 2x – 1 , it is in the form of y = mx + c,
Given to which figure has an area of 81 square units
substitute y  = 81, we have
y = 2x – 1
81 = 2x – 1
2x = 82
x = 41
So, For figure has an area of 81 square units is 41.

b. As shown above , we know that ,
figure 1 = 1 square unit
figure 2 =4 square units
figure 3 = 9 square unit and so on
Here we have a pattern of power of its own number,
So, the Equation  will be y = x²
Given to which figure has an area of 81 square units
substitute y  = 81, we have
x  = 9
So, For figure has an area of 81 square units is 9.

EXPLORATION 2

Using a Graph
Work with a partner. Use a graph to test the truth of each statement. If the statement is true, write an equation that shows how to obtain one measurement from the other.

a. “You can find the horsepower of a race-car car engine if you know its volume in cubic inches”

b. “You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches.”

Answer: a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600)
We can not find the horsepower of a race-car car engine if you know its volume in cubic inches
b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) ,
Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches

Explanation:
a. Given ordered pairs are (200 , 375) , (350 , 650) , (350 , 250) , (500 , 600)
We can not find the horsepower of a race-car car engine if you know its volume in cubic inches

b. Given ordered pairs are (100 , 1640) , (200 , 3280) , (300 , 4920) ,
Yes, You can find the volume of a race-car engine in cubic centimeters if you know its volume in cubic inches

Try It

Question 1.
Write a function rule for “The output is one-fourth of the input.”
Answer:  y = \(\frac{x}{4}\)

Explanation:
Let us say x is input and y is output , then
The output is one-fourth of the input, will be ,
y = \(\frac{x}{4}\).

Find the value of y when x = 5.
Question 2.
y = 4x – 1
Answer: y = 19.

Explanation:
Given, y = 4x – 1
substitute x = 5 , we get
y = 4(5) – 1
y = 20 – 1 = 19
So, y = 19.

Question 3.
y = 10x
Answer: y = 50

Explanation:
Given, y =10x
substitute x = 5 , we get
y = 10(5)
y = 50
So, y = 50.

Question 4.
y = 7 – 3x
Answer: y = -8.

Explanation:
Given, y = 7 – 3x
substitute x = 5 , we get
y = 7 – 3(5)
y = 7 – 15 = -8
So, y = -8.

Graph the function.
Question 5.
y = x + 1
Answer:

Explanation:
Given , y = x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3)
The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line .

Question 6.
y = – 3x
Answer:

Explanation:
Given , y = – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = -3(1) = -3 . co-ordinates are (1 , -3)
if x = 2 , then y = -3(2) = -6 , co-ordinates are (2 , -6)
if x = 3 , then y = -3(3) = -9 , co-ordinates are (3 , -9)
The co-ordinates (0 , 0) , (1 , -3) , (2 , -6) ,(3 , -9) form a straight line .

Question 7.
y = 3x + 2
Answer:

Explanation:
Given , y = 3x + 2  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =3(0) + 2 = 2 . co-ordinates are (0 , 2)
if x = 1 , then y = 3(1) + 2= 5 . co-ordinates are (1 , 5)
if x = 2 , then y =3(2) + 2 = 7 , co-ordinates are (2 , 7)
The co-ordinates (0 , 2) , (1 , 5) , (2 , 7) form a straight line .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

WRITING FUNCTION RULES Write a function rule for the statement.
Question 8.
The output is three times the input.
Answer: y = 3x

Explanation:
Let us say x is input and y is output , then
The output is three times the input. will be ,
So , y = 3x .

Question 9.
The output is eight more than one-seventh of the input.
Answer: y = 8 + \(\frac{x}{7}\) .

Explanation:
Let us say x is input and y is output , then
The output is eight more than one-seventh of the input., will be ,
So, y = 8 + \(\frac{x}{7}\) .

EVALUATING A FUNCTION Find the value of y when x = 5.
Question 10.
y = 6x
Answer: y = 30

Explanation:
Given, y = 6x
substitute x = 5 , we get
y = 6(5) =30
So, y = 30

Question 11.
y = 11 – x
Answer: y = 6

Explanation:
Given, y = 11 – x
substitute x = 5 , we get
y = 11 – 5 = 6
So, y = 6.

Question 12.
y = \(\frac{1}{5}\)x + 1
Answer:  y = 2.

Explanation:
Given, y = \(\frac{1}{5}\)x + 1
substitute x = 5 , we get
y = \(\frac{x}{5}\) + 1
y= \(\frac{5}{5}\) + 1
y = 1 + 1 = 2
So, y = 2 .

GRAPHING A FUNCTION Graph the function.
Question 13.
y = – 2x
Answer:

Explanation:
Given , y = – 2x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 2(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 2(1)= -2 . co-ordinates are (1 , -2)
if x = 2 , then y =- 2(2) = -4 , co-ordinates are (2 , -4)
if x = 3 , then y =- 2(3) = -6 , co-ordinates are (3 , -6)
The co-ordinates (0 , 0) , (1 , -2) , (2 , -4) , (3 , -6) form a straight line .

Question 14.
y = x – 3
Answer:

Explanation:
Given , y = x – 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 – 3 = -3 . co-ordinates are (0 , -3)
if x = 1 , then y = 1 – 3= -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 2 – 3 = -1 , co-ordinates are (2 , -1)
if x = 3 , then y = 3 – 3 = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , -3) , (1 , -2) , (2 , -1) , (3 , 0) form a straight line .

Question 15.
y = 9 – 3x
Answer: 

Explanation:
Given , y = 9 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 9 – 3(0) = 9 . co-ordinates are (0 , 9)
if x = 1 , then y = 9 – 3(1) = 6 . co-ordinates are (1 , 6)
if x = 2 , then y = 9 – 3(2) = 3 , co-ordinates are (2 , 3)
if x = 3 , then y = 9 – 3(3) = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , 9) , (1 , 6) , (2 , 3) , (3 , 0) form a straight line .

Question 16.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

Answer: As mentioned in the explanation below  a & d , b & c are different .

Explanation:
Given ,
a. what output is 4 more than twice the input 3?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 4 + 2(3) = 10.
b. What output is twice the sum of the input 3 and 4?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 2( 3 + 4 ) = 14.
c. what output is the sum of 2 times the input 3 and 4?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 2( 3 + 4 ) = 14.
d. what output is 4 increased by twice the input 3?
Let us say that , y is output and x is input and given as 3 ,
then, we have y = 4 + 2(3) = 10.

So, a & d , b & c are different .

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 17.
The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people. How many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people? Justify your answer using a graph.

Answer: So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people

Explanation:
Given, The World Health Organization(WHO) suggests having 23 health-care workers for every 10,000 people.
we need to find how many health-care workers are needed to meet the WHO suggestion for a population of 250,000 people,
For every 10,000 people we have 23 care takers
Then for 250,000 people we have
\(\frac{23 × 250,000}{10,000}\)
= 23 × 25
= 575
So, 575 health-care workers are needed to meet the WHO suggestion for a population of 250,000 people

Question 18.
DIG DEEPER!
A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned. The fuel economy of the truck is 18 miles per gallon. Write and graph a function that describes the relationship between carbon dioxide produced and distance traveled.
Answer: y = 22x + 18  is the linear equation

Explanation:
Given, A truck produces 22 pounds of carbon dioxide for every gallon of diesel fuel burned.
The fuel economy of the truck is 18 miles per gallon.
So, we have y = 22x + 18 is in the form of y = mx +c
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 22(0) + 18 = 18 . co-ordinates are (0 , 18)
if x = 1 , then y =22(1) + 18= 40  . co-ordinates are (1 , 40)
if x = 2 , then y =22(2) + 18 = 62 , co-ordinates are (2 , 62)
if x = 3 , then y =22(3) + 18 = 84  , co-ordinates are (3 , 84)
The co-ordinates (0 , 18) , (1 , 40) , (2 , 62) , (3 , 84) form a straight line .
The graph is

Representations of Functions Homework & Practice 7.2

Review & Refresh

Determine whether the relation is a function. 
Question 1.

Answer:  The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 2.

Answer:  The relation is a function .

Explanation:
Each input has exactly one output ,
So , The relation is a function .

Question 3.

Answer: The relation is not a function .

Explanation:
The each input has  more than two outputs , That is one input has multiple number of outputs.
Here , input 2 has two outputs which are 0 and -4 .
So , The relation is not a function .

Find the slope of the line.
Question 4.

Answer: slope = 1.

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -2 to -4 =2
change in x or ▲x is change from 1 to 3 = 2 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{2}{2}\)
slope = 1.

Question 5.

Answer: slope = \(\frac{5}{2}\) .

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -4 to 1 = 5
change in x or ▲x is change from -1 to -3 = 2 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{5}{2}\) .

Question 6.

Answer:  slope = \(\frac{1}{3}\) .

Explanation:
By using the slope equation , we know that
Slope = \(\frac{change in y}{change in x}\) or
slope = \(\frac{▲y}{▲x}\)
From the graph we know that change in y or ▲y is change from -4 to -3 = 1
change in x or ▲x is change from 1 to 4 = 3 ,
So, slope = \(\frac{▲y}{▲x}\)
slope = \(\frac{1}{3}\) .

Concepts, Skills, & Problem Solving

USING A GRAPH Use a graph to test the truth of the statement. If the statement is true, write an equation that shows how to obtain one measurement from the other measurement. (See Exploration 2, p. 281.)

Question 7.
“You can find the weight of a cell phone in ounces if you know its screen size in inches.”

Answer: we can does not find the weight of a cell phone in ounces if you know its screen size in inches.

From the given table , Ordered pairs are (4 , 4) , (4.7 , 4.8) , (5 , 4.8) , (5.5 , 6.4)
First find the slope m of the line containing the two given points (4, 4) and (4.7, 4.8)
m = (y₂-y₁) / (x₂-x₁)
m= (4.8 – 4) / (4.7 – 4)
m = 0.8/0.7 .
So, we can does not find the weight of a cell phone in ounces if you know its screen size in inches.

Question 8.
“You can find the age of a child in years if you know the age of the child in months.”

Answer: YES, y = 0.08x + 0.04 is a linear equations

Explanation:
From the given table , Ordered pairs are (9 , 0.75) , (12 , 1) , (15 , 1.25) , (24 , 2)
First find the slope m of the line containing the two given points (12 ,1) and (24, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 1) / (24 – 12)
m = 1/12
m = 0.08.
substitute the slope in the (12 ,1) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 1 = 0.08(x – 12)
y –1 = 0.08x – 0.96
y = 0.08x –0.96 + 1
y =0.08 x + 0.04
So, y = 0.08x + 0.04 is a linear equation

WRITING FUNCTION RULES Write a function rule for the statement.
Question 9.
The output is half of the input.
Answer: y = \(\frac{x}{2}\).

Explanation:
Let us say x is input and y is output , then
The output is half of the input, will be ,
y = \(\frac{x}{2}\).

Question 10.
The output is eleven more than the input.
Answer: y = x + 11

Explanation:
Let us say x is input and y is output , then
The output is eleven more than the input, will be ,
y = x + 11

Question 11.
The output is three less than the input.
Answer: y = x – 3

Explanation:
Let us say x is input and y is output , then
The output is three less than the input, will be ,
y = x – 3

Question 12.
The output is the cube of the input.
Answer: y = x³

Explanation:
Let us say x is input and y is output , then
The output is the cube of the input, will be ,
y = x³

Question 13.
The output is six times the input.
Answer: y = 6x

Explanation:
Let us say x is input and y is output , then
The output is six times the input, will be ,
y = 6x

Question 14.
The output is one more than twice the input.
Answer: y = 2x + 1

Explanation:
Let us say x is input and y is output , then
The output is one more than twice the input, will be ,
y = 2x + 1

EVALUATING A FUNCTION Find the value of y for the given value of x.
Question 15.
y = x + 5; x = 3
Answer: y = 8

Explanation:
Given, y = x + 5
substitute x = 3 , we get
y = 3 + 5
So, y = 8.

Question 16.
y = 7x; x = – 5
Answer:  y = -35.

Explanation:
Given, y = 7x
substitute x = -5 , we get
y = 7(-5)
So, y = -35.

Question 17.
y = 1 – 2x; x = 9
Answer: y = -17

Explanation:
Given, y = 1 – 2x
substitute x = 9 , we get
y = 1 – 2(9)
y = 1 – 18
So, y = -17.

Question 18.
y = 3x + 2; x = 0.5
Answer: y = 5.5

Explanation:
Given, y = 3x + 2
substitute x = 0.5 , we get
y = 3(0.5) + 2
y = 3.5 + 2
So, y = 5.5 .

Question 19.
y = 2x³; x = 3
Answer: y = 54

Explanation:
Given, y = 2x³
substitute x = 3 , we get
y = 2(3)³
y = 2 × 27 = 54
So, y = 54.

Question 20.
y = \(\frac{x}{2}\) + 9; x = – 12
Answer: y = 3

Explanation:
Given, y = \(\frac{x}{2}\) + 9
substitute x = -12 , we get
y = \(\frac{-12}{2}\) + 9
y = -6 + 9
So, y = 3 .

GRAPHING A FUNCTION Graph the function.
Question 21.
y = x + 4
Answer: 

Explanation:
Given , y = x + 4  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 4 = 4 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 4 = 5 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 4 = 6 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 4 = 7 , co-ordinates are (3 , 7)
The co-ordinates (0 , 4) , (1 , 5) , (2 , 6) , (3 , 7) form a straight line .

Question 22.
y = 2x
Answer:

Explanation:
Given , y = 2x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2(1) = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2(2) = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = 2(3) = 6 , co-ordinates are (3 , 6)
The co-ordinates (0 , 0) , (1 , 2) , (2 , 4) , (3 , 6) form a straight line .

Question 23.
y = – 5x + 3
Answer:

Explanation:
Given , y = – 5x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = – 5(1) + 3 = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = – 5(2) + 3 = -7 , co-ordinates are (2 , -7)
if x = 3 , then y = – 5(3) + 3 = -12 , co-ordinates are (3 , -12)
The co-ordinates (0 , 3) , (1 , -2) , (2 , -7) , (3 , -12) form a straight line .

Question 24.
y = \(\frac{x}{4}\)
Answer:

Explanation:
Given , y = \(\frac{x}{4}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{0}{4}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{1}{4}\) = 0.25 . co-ordinates are (1 , 0.25)
if x = 2 , then y = \(\frac{2}{4}\) = 0.5 , co-ordinates are (2 , 0.5)
if x = 3 , then y = \(\frac{3}{4}\) = 0.75 , co-ordinates are (3 , 0.75)
The co-ordinates (0 , 0) , (1 , 0.25) , (2 , 0.5) , (3 , 0.75) form a straight line .

Question 25.
y = \(\frac{3}{2}\)x + 1
Answer:

Explanation:
Given , y = \(\frac{3}{2}\)x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =\(\frac{3}{2}\)(0) + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = \(\frac{3}{2}\)(1) + 1= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = \(\frac{3}{2}\)(2) + 1 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = \(\frac{3}{2}\)(3) + 1 = 5.5 , co-ordinates are (3 , 5.5)
The co-ordinates (0 , 1) , (1 , 2.5) , (2 , 4) , (3 , 5.5) form a straight line .

Question 26.
y = 1 + 0.5x
Answer:

Explanation:
Given , y = 1 + 0.5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 + 0.5(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 0.5(1) = 1.5 . co-ordinates are (1 , 1.5)
if x = 2 , then y = 1 + 0.5(2) = 2 , co-ordinates are (2 , 2)
if x = 3 , then y = 1 + 0.5(3) = 2.5 , co-ordinates are (3 , 2.5)
The co-ordinates (0 , 1) , (1 , 1.5) , (2 , 2) , (3 , 2.5)  form a straight line .

MATCHING Match the graph with the function it represents.
A. y = \(\frac{x}{3}\)
B. y = x + 1
C. y = – 2x + 6
Question 27.

Answer:  B. y = x + 1.

Explanation:
Given , y = x + 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 1 = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 + 1 = 2 . co-ordinates are (1 , 2)
if x = 2 , then y = 2 + 1 = 3 , co-ordinates are (2 , 3)
The co-ordinates (0 , 1) , (1 , 2) , (2 , 3) form a straight line .

Question 28.

Answer: c. y = – 2x + 6

Explanation:
Given , y = – 2x + 6  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = – 2(0) + 6 = 6 . co-ordinates are (0 , 6)
if x = 1 , then y = – 2(1) + 6 = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = – 2(2) + 6 = 2 , co-ordinates are (2 , 2)
if x = 3 , then y = – 2(3) + 6 = 0 , co-ordinates are (3 , 0)
The co-ordinates (0 , 6) , (1 , 4) , (2 , 2) , (3 , 0) form a straight line .

Question 29.

Answer: A. y = \(\frac{x}{3}\)

Explanation:
Given , y =  \(\frac{x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =\(\frac{0}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{1}{3}\)= 0.3 . co-ordinates are (1 , 0.3)
if x = 2 , then y = \(\frac{2}{3}\)= 0.6 , co-ordinates are (2 , 0.6)
The co-ordinates (0 , 0) , (1 , 0.3) , (2 , 0.6) form a straight line .

Question 30.
YOU BE THE TEACHER
Your friend graphs the function represented by the input-output table. Is your friend correct? Explain your reasoning.

Answer: Yes , He is correct

Explanation:

Ordered pairs are (-1 , -4) , (1 , -2) , (3 ,0) , (5 , 2)
these points form a straight line when graphed.
Yes , He is correct

Question 31.
MODELING REAL LIFE
A dolphin eats 30 pounds of fish per day.

a. Write and graph a function that relates the number p of pounds of fish that a dolphin eats in d days.
b. How many total pounds of fish does a dolphin eat in 30 days?
Answer:

Explanation:
a. Given , A dolphin eats 30 pounds of fish per day.
by each passing day eating fish is increased by the day passes .
So, y = 30x is the function,
The graph represents the function as

b. Given , A dolphin eats 30 pounds of fish per day.
then for 30 days ,
30 × 30 = 900 pounds
So, A dolphin eats 900 pounds of fish in 30 days

Question 32.
MODELING REAL LIFE
You fill a fish tank with 55 gallons of water on Saturday. The water evaporates at a rate of 1.5 gallons per day. You plan to add water when the tank reaches 49 gallons. When will you add water? Justify your answer.
Answer: As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday.

Explanation:
Given data ,, implies that slope of the function m = -1.5
The y intercept b= 55,
Then the equation  will be y = 55 – 1.5x
Given , You plan to add water when the tank reaches 49 gallons.
determine x for y = 49 ,
So, 49 = 55 – 1.5x ,
1.5x = 55 – 49
1.5x = 6
x = \(\frac{6}{1.5}\)
x = 4.

As the action starts on Saturday , the tank will reach 49 gallons after 4 days , That is on Wednesday.

USING AN EQUATION Find the value of x for the given value of y.
Question 33.
y = 5x – 7; y = – 22
Answer: x = -3

Explanation:
Given, y = 5x – 7
x = \(\frac{y + 7}{5}\)
substitute y = -22 , we get
x = \(\frac{-22 + 7}{5}\)
x = \(\frac{- 15}{5}\)
x = -3
So, x = -3 .

Question 34.
y = 9 – 7x; y = 37
Answer: x = -4

Explanation:
Given, y = 9 – 7x
x = \(\frac{9 – y}{7}\)
substitute y = 37 , we get
x = \(\frac{9 – 37}{7}\)
x = \(\frac{- 28}{7}\)
x = -4
So, x = -4 .

Question 35.
y = \(\frac{x}{4}\) – 7; y = 2
Answer: x = 36

Explanation:
Given, y = \(\frac{x}{4}\) – 7
x = 4( y + 7)
substitute y = 2 , we get
x = 4( 2 + 7)
x = 4(9)
x = 36
So, x = 36 .

Question 36.
PROBLEM SOLVING
You decide to make and sell bracelets. The cost of your materials is $84.00. You charge $3.50 for each bracelet.

a. P Write a function that represents the profit for selling b bracelets.
b. Which variable is independent? dependent? Explain.
c. You will break even when the cost of your materials equals your income. How many bracelets must you sell to break even?
Answer: a. A function that represents the profit for selling b bracelets is p = 3.5b – 84.
b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable.
c. To break even you must sell 24  bracelets.

Explanation:
a. Given , The cost of your materials is $84.00. You charge $3.50 for each bracelet,
Let p be the profit , b be the number of bracelets sold,
So, profit = income – cost .
p = 3.5b – 84.
Thus , A function that represents the profit for selling b bracelets is p = 3.5b – 84.

b. Here , the profit depends on the number of bracelets sold , b is the independent variable and p is the dependent variable.

c. set the income expression from part a equal to the cost of 84 and solve for b ,
So, income = cost .
3.5b = 84 ,
b = \(\frac{84}{3.5}\)
b = 24.

To break even you must sell 24  bracelets.

Question 37.
MODELING REAL LIFE
A furniture store is having a sale where everything is 40% off.
a. Write and graph a function that represents the amount of discount on an item at regular price.
b. You buy a bookshelf that has a regular price of $85. What is the sale price of the bookshelf?
Answer: a. The function is y = 0.4x and the graph is given below.
b. The sale price of the bookshelf s $51.

Explanation:
a. A function that represents the amount of discount on an item at regular price is ,
Given , 40% = 0.4 ,
To find the percent of the number , we should multiply the number by the percent in the decimal form ,
so, the equation is d = 0.4p ,
let us convert it in to a function form , y = 0.4x
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0.4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 0.4(1)= 0.4 . co-ordinates are (1 , 0.4)
if x = 2 , then y =0.4(2) = 0.8 , co-ordinates are (2 , 0.8)
if x = 3 , then y = 0.4(3) = 1.2 , co-ordinates are (3 , 1.2)
The co-ordinates (0 , 0) , (1 , 0.4) , (2 , 0.8) , (3 , 1.2) form a straight line .
The graph is
b. Given , You buy a bookshelf that has a regular price of $85.
The sale price of the bookshelf is ,
substituting the given price in p = 85 ,
it will be the discount d = 0.4 (85) = 34
Then the sale price is $85 – $34 = $51.

So, The sale price of the bookshelf s $51.

Question 38.
REASONING
You want to take a two-hour air boat tour. Which is a better deal, Snake Tours or Gator Tours? Use functions to justify your answer.

Answer: By using functions , $50 > $40 , So, Gator tours are cheaper than the snake tours .

Explanation:
Given , You want to take a two-hour air boat tour.
Let x be the hours of  air boat tour and y be the cost of air boat tour ,
Snake tours , y = 25x
putt x = 2 ,
So , y = 25 (2) = 50 .
y = 50.

Gator tour , y = 35 + \(\frac{5}{2}\)x
Put x = 2 ,
So, y = 35 + \(\frac{5}{2}\) x
y = 35 + 2.5x
y = 35 + 2.5 (2)
y = 35 + 5
y = 40 .

Finally $50 > $40 , So, Gator tours are cheaper than the snake tours

Question 39.
REASONING
The graph of a function is a line that passes through the points (3, 2), (5, 8), and (8, y). What is the value of y?
Answer: The value of y is 17 , so, The third given point is (8, 17)

Explanation:
First find the slope m of the line containing the two given points (3,2) and (5,8)
m = (y₂-y₁) / (x₂-x₁)
m= (8 – 2) / (5 – 3)
m = 6 / 2
m = 3
Then use the slope and one of the given points (3,2) to find the y-intercept
y = mx +
2 = 3(3) + b
2 = 9 + b
-7 = b
The equation is   y = 3x -7
Then find the third point (8, y) by replacing x by 8
y = 3x -7
y = 3(8) -7
y = 24 -7
y = 17

Question 40.
CRITICAL THINKING
Make a table where the independent variable is the side length of a square and the dependent variable is the perimeter. Make a second table where the independent variable is the side length of a square and the dependent variable is the area. Graph both functions in the same coordinate plane. Compare the functions.
Answer: The graph for the perimeter is linear , The graph for the Area is Quadratic .

Explanation:
Let us say , s be the side length of the square ,
Then the perimeter is P = 4s ,
The function will be y= 4x,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 4(1) = 4 . co-ordinates are (1 , 4)
if x = 2 , then y = 4(2) =8 , co-ordinates are (2 , 8)
if x = 3 , then y = 4(3) = 0 , co-ordinates are (3 , 12)
The co-ordinates (0 , 0) , (1 , 4) , (2 ,8) , (3 , 12) form a straight line .

Table will be ,

Let us say , s be the side length of the square ,
Then the Area is A = s² ,
The function will be y=x²,
we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0² = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 1² = 1 . co-ordinates are (1 , 1)
if x = 2 , then y = 2² =4 , co-ordinates are (2 , 4)
if x = 3 , then y = 3² = 9 , co-ordinates are (3 , 9)
The co-ordinates (0 , 0) , (1 , 1) , (2 ,4) , (3 , 9) form a straight line .

Second table is
Then the graph is 
The graph for the perimeter is linear , The graph for the Area is Quadratic .

Question 41.
PUZZLE
The blocks that form the diagonals of each square are shaded. Each block has an area of one square unit. Find the “green area” of Square 20. Find the “green area” of Square 21. Explain your reasoning.

Answer:  The green area of the Square 20 is 46 square units and The green area of the Square 21 is 48 square units.

Explanation:
Given , Each block has an area of one square unit,
Square 1 has  the diagonals of each square are shaded. the “green area” is 3 + 3 = 6 square units ,
Square 2 has  the diagonals of each square are shaded. the “green area” is 4 + 4 = 8 square units ,
Square 3 has  the diagonals of each square are shaded. the “green area” is 5 + 5 = 10 square units ,
Square 4 has  the diagonals of each square are shaded. the “green area” is 6 + 6 = 12 square units,
Square 5 has  the diagonals of each square are shaded. the “green area” is 7 + 7 = 14 square units ,
Here , The number of squares are increasing by one block with the square numbers.
So for the , Square 20 has  the diagonals of each square are shaded. the “green area” is 23 + 23 = 46 square units,
And Square 21 has  the diagonals of each square are shaded. the “green area” is 24 + 24 = 48 square units.

Lesson 7.3 Linear Functions

EXPLORATION 1

Writing and Graphing Functions
Work with a partner. Each table shows a familiar pattern from geometry.

• Determine what the variables x and y represent. Then write a function rule that relates y to x.
• Is the function a linear function? Explain your reasoning.

Answer: All of them are explained below

Explanation:
The variables x and y represents a rectangle
a. From the given table , Ordered pairs are (1 , 10) , (2 , 12) , (3 , 14) , (4 , 16)
First find the slope m of the line containing the two given points (1 ,-1) and (2, -2)
m = (y₂-y₁) / (x₂-x₁)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
substitute the slope in the (1 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = -1/2 ( x –1)
2(y – 10) = -x  + 1
2y – 20 = -x+ 1
2y = -x  + 21
y = \(\frac{-1}{2}\) (x – 21)
So ,  y = \(\frac{-1}{2}\) (x – 21) is linear function.

b. The variables x and y represent a circle
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y₂-y₁) / (x₂-x₁)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

c. The variables x and y represents a trapezoid
a. From the given table , Ordered pairs are (1 , 5) , (2 , 6) , (3 , 7) , (4 , 8)
First find the slope m of the line containing the two given points (1 ,5) and (2, 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 5) / (2 – 1)
m = 1 .
substitute the slope in the (1 ,5) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 5 = 1(x – 1)
y – 5 = x – 1
y = x – 1 + 5
y = x + 4
So, y = x + 4 is a linear equation

d. The variables x and y represents a cube
a. From the given table , Ordered pairs are (1 , 28) , (2 , 40) , (3 , 52) , (4 , 64)
First find the slope m of the line containing the two given points (1 ,28) and (2, 40)
m = (y₂-y₁) / (x₂-x₁)
m= (40 – 28) / (2 – 1)
m = 12 .
substitute the slope in the (1 ,28) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 28 = 12(x – 1)
y – 28 = 12x – 12
y = 12x – 12 + 28
y = 12x + 16
So, y = 12x + 16 is a linear equation.

Try It

Question 1.
Use the graph to write a linear function that relates y to x.

Answer: the linear function is y = \(\frac{-1}{2}\)x -1.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , 1) , (-2 , 0 ) , (0 , -1) , ( 2, -2 )
First find the slope m of the line containing the two given points (0 ,-1) and (2, -2)
m = (y₂-y₁) / (x₂-x₁)
m= (-2 – (-1)) / (2 – 0)
m = -1 / 2 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = \(\frac{-1}{2}\)x -1.

Question 2.
Use the table to write a linear function that relates y to x.

Answer: the linear function is y = (0)x + 2.

Explanation:
Ordered pairs are (-2 , 2) , (-1 , 2) , (0 , 2) , (1 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,2) and (1, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 2) / (1 – 0)
m = 0
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = (0)x + 2.

Question 3.
WHAT IF?
The rate of descent doubles. Repeat parts (a) and (b).
Answer: a. the linear function is y = -1x + 65.
b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Explanation:
a. From the Given table , The rate of descents is 5
If it doubles , then The rate of descents is 10.
The the ordered pairs will be (0 , 65) , (10 ,55) , (20 , 45) .
First find the slope m of the line containing the two given points (0 ,65) and (10, 55)
m = (y₂-y₁) / (x₂-x₁)
m= (55 – 65) / (10 – 0)
m = -10 / 10
m = -1
Because the line crosses the y axis at ( 0, 65 ) , The y intercept is 65.
So , the linear function is y = -1x + 65.

b. The slope indicates that the height decreases 1000 feet per minute.
The y intercept indicates that the descent begins at a cruising altitude of 65,000 feet.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.
WRITING A LINEAR FUNCTION
Use the graph to write a linear function that relates y to x.

Answer:  The linear function is y = -4x -2 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 6) , (-1 , 2 ) , (0 , -2) , ( 1, -6 )
First find the slope m of the line containing the two given points (0 ,-2) and (1, -6)
m = (y₂-y₁) / (x₂-x₁)
m= (-6 – (-2)) / (1 – 0)
m = -4 .
Because the line crosses the y axis at ( 0, -2) , The y intercept is -2.
So , the linear function is y = -4x -2 .

Question 5.
INTERPRETING A LINEAR FUNCTION
The table shows the revenue R (in millions of dollars) of a company when it spends A (in millions of dollars) on advertising.

a. Write and graph a linear function that relates R to A.
b. Interpret the slope and the y-intercept.
Answer:  a. The linear function is y = 2x + 2. and the graph is shown below
b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Explanation:
a. From the given table ,
The the ordered pairs will be (0 , 2) , (2 ,6) , (4 , 10) , (6 , 14) , (8 ,18) .
The graph is
First find the slope m of the line containing the two given points (0 ,2) and (2, 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 2) / (2 – 0)
m = 4 / 2
m = 2
Because the line crosses the y axis at ( 0, 2 ) , The y intercept is 2.
So , the linear function is y = 2x + 2.

b. The slope indicates that the increasing in the amount of spending on advertising by 2 million dollars
The y intercept indicates that the Revenue begins to increasing from the 2 million dollars.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.
Manager A earns $15 per hour and receives a $50 bonus. The graph shows the earnings of Manager B. (a) Which manager has a greater hourly wage? (b) After how many hours does Manager B earn more money than Manager A?

Answer: a. Manager B has the greater hourly wage than Manager A .
b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Explanation:
a. Manager A earns $15 per hour and receives a $50 bonus.
The ordered pairs will be  (0 , 0) , (1 , 15) , (2 , 30) , (3 , 45)
The graph shows the earnings of Manager B.
Ordered pairs from the graph are  (0 , 0) , (1 , 25) , (2 , 50) , (3 , 75)
So, Manager B has the greater hourly wage than Manager A .

b. As manager A receives a $50 bonus , Manager B has to work an hour extra to earn more money than Manager A .

Question 7.
Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day. The table shows the amount (in gigabytes) of data that your friend has left days after the start of each month. Who runs out of data first? Justify your answer.

Answer:  you will be run out of data first

Explanation:
a. Given , Each month, you start with 2 gigabytes of data and use 0.08 gigabyte per day.
Let x be the number of days and y be the total data in gigabytes.
So, y = -0.08x + 2 ,
You will be out of data if , -0.08x + 2 = 0 ,
-0.08x + 2 = 0
2 = 0.08x
x = \(\frac{2}{0.08}\)
x = 25.
Hence ,you will be run out of data in 25 days.
b. Daily data usage for the friend will be given by the slope of the graph.
The the ordered pairs will be (0 , 3) , (7 ,2.3) , (14 , 1.6) .
First find the slope m of the line containing the two given points (7 ,2.3) and (14, 1.6)
m = (y₂-y₁) / (x₂-x₁)
m= (1.6 – 2.3) / (14 – 7)
m = -0.7 / 7
m = -0.1
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = -0.1x + 3.
Your friend will be out of data if ,
-0.1x + 3 = 0
3 = 0.1x
x = \(\frac{3}{0.1}\)
x = 30 .
Hence ,Friend will be run out of data in 30 days

So , you will be run out of data first

Linear Functions Homework & Practice 7.3

Review & Refresh

Write a function rule for the statement. Then graph the function.
Question 1.
The output is ten less than the input.
Answer: y = x – 10.

Explanation:
Let us say x is input and y is output , then
The output is ten less than the input, will be ,
y = x – 10.

Question 2.
The output is one-third of the input.
Answer: y = \(\frac{x}{3}\)

Explanation:
Let us say x is input and y is output , then
The output is one-third of the input, will be ,
y = \(\frac{x}{3}\) .

Solve the system.
Question 3.
y = x + 5
y = – 3x + 1
Answer: X = 0 , Y = 5

Explanation:
$Y=-3X+5$ ——————-(1)
$Y=X+5$ ——————(2)
Substitute $Y=X+5$ in equation (1)
X+5=−3X+5
Solve it for X
X+3X=5−5
4X=0
X=0/4=0
X = 0
Substitute $X=0$ in equation (1)
Y=0+5
Y=5

Question 4.
x + y = – 4
6x + 2y = 4
Answer:  X = 3 , $\mathrm{Y=\; -7\; .}$

Explanation:
$\mathrm{2Y}=-6X+4$ ——————-(1)
$Y=\; –\mathrm{X-4}$ ——————(2)
Substitute $Y=\; –\mathrm{X-4}$ in equation (1)
$\mathrm{2Y}=-6X+4$
2 ( $–\mathrm{X\; –\; 4\; )\; =\; -6X\; +\; 4\; -2X\; –\; 8\; =\; -6X\; +\; 4\; 6X\; -2X\; =\; 8\; +\; 4\; 4X\; =\; 12\; X\; =\; 3\; Substitute X=3 in\; equation\; (2)\; Y=–\; 3\; –\; 4\; Y=\; -7\; .}$

Question 5.
– 4x + 3y = 14
y = 2x + 8
Answer:  X = -5 , $Y\; =\; -2\; .$

Explanation:
$\mathrm{3Y}=\; 4X+14$ ——————-(1)
Y = 2X + 8 ——————(2)
Substitute $Y=\; 2X\; +\; 8$ in equation (1)
$\mathrm{3Y}=\; 4X+14$
3($2X\; +\; 8)\; =\; 4X+14\; 6X\; +\; 24\; =\; 4X\; +\; 14\; 6X\; –\; 4X\; =\; 14\; –\; 24\; 2X\; =\; -10\; X\; =\; -5\; Substitute X=\; -5 in\; equation\; (2)\; Y=\; 2(-5)\; +\; 8\; Y=\; -10\; +\; 8\; Y\; =\; -2.$

Concepts, Skills, &Problem Solving

WRITING AND GRAPHING FUNCTIONS The table shows a familiar pattern from geometry. (a) Determine what the variables x and y represent. Then write a function rule that relates y to x. (b) Is the function a linear function? Explain your reasoning. (See Exploration 1, p. 289.)
Question 6.

Answer: a. The variables x and y represent a right angle triangle
b. y = 2x  is linear function.

Explanation:
In order to write the function we have to write the ordered pairs
Ordered pairs are  (1 , 2) , (2 , 4) ,  (3 , 6 ) , (4 , 8), (5 , 10 ) .
a. the variables x and y represent a right angle triangle
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 2) , (2 , 4)
m = (y₂-y₁) / (x₂-x₁)
m= (4 – 2) / (2– 1)
m = 2/1
m = 2
b. substitute the slope in the (2 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4 = 2 ( x – 2)
y – 4 = 2x – 4
y = 2x – 4 + 4
y = 2x
So ,  y = 2x  is linear function.

Given side of triangle is 4 then x= 4/2 = 2
x = 2 and y = 4.

Question 7.

Answer: y = 3.14x  is linear function. and The variables x and y represent a circle

Explanation:
Ordered pairs are (1 , 3.14 ) , (2 , 6.28) , (3 , 9.42) , (4 , 12.5 )
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3.14 ) , (2 , 6.28)
m = (y₂-y₁) / (x₂-x₁)
m= (6.28 –3.14) / (2– 1)
m = 3.14/1
m = 3.14
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3.14 =3.14 ( x –1)
y – 3.14 = 3.14x – 3.14
y = 3.14x – 3.14 + 3.14
y = 3.14x
So ,  y = 3.14x  is linear function.
Where x is the diameter of the circle.

WRITING LINEAR FUNCTIONS Use the graph or table to write a linear function that relates y to x.
Question 8.

Answer: The linear function is y = \(\frac{4}{3}\)x +2

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-3 , -2) , (0 , 2 ) , (3 , 6) , ( 6, 10 )
First find the slope m of the line containing the two given points (3 ,6) and (6, 10)
m = (y₂-y₁) / (x₂-x₁)
m= (10 – 6) / (6 – 3)
m = 4/3 .
Because the line crosses the y axis at ( 0, 2) , The y intercept is 2.
So , the linear function is y = \(\frac{4}{3}\)x +2 .

Question 9.

Answer: The linear function is y = (0)x +3 .

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , 3) , (-1 , 3 ) , (0 , 3) , ( 1, 3 ) , (2 , 3)
First find the slope m of the line containing the two given points (1 ,3) and (2, 3)
m = (y₂-y₁) / (x₂-x₁)
m= (3 – 3) / (2 – 1)
m = 0 .
Because the line crosses the y axis at ( 0, 3) , The y intercept is 3.
So , the linear function is y = (0)x +3 .

Question 10.

Answer: The linear function is y = \(\frac{-1}{4}\)x + 0.

Explanation:
Ordered pairs are (-8 , 2) , (-4 , 1) , (0 , 0) , (4 , -1)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (-8 ,2) and (-4, 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 2) / (-4 – (-8))
m = -1/4
Because the line crosses the y axis at ( 0, 0 ) , The y intercept is 0.
So , the linear function is y = \(\frac{-1}{4}\)x + 0.

Question 11.

Answer: The linear function is y = \(\frac{2}{3}\)x + 5.

Explanation:
Ordered pairs are (-3 , 3) , (0 , 5) , (3 , 7) , (6 , 9)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 ,7) and (6, 9)
m = (y₂-y₁) / (x₂-x₁)
m= (9 – 7) / (6 – 3)
m = 2/3
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y = \(\frac{2}{3}\)x + 5.

Question 12.
INTERPRETING A LINEAR FUNCTION
The table shows the length y (in inches) of a person’s hair after x months.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
Answer: a. The linear function is y = 0.5x + 11.
b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Explanation:
a. Given ,
The ordered pairs will be (0 , 11) , (3 ,12.5) , (6 , 14) .
The graph is
First find the slope m of the line containing the two given points (3 ,12.5) and (6 , 14)
m = (y₂-y₁) / (x₂-x₁)
m= (14 – 12.5) / (6 – 3)
m = 1.5 / 3
m = 0.5
Because the line crosses the y axis at ( 0, 11 ) , The y intercept is 11.
So , the linear function is y = 0.5x + 11.

b. The slope indicates that the increasing in the hair length
The y intercept indicates that the increasing in hair length by time.

Question 13.
INTERPRETING A LINEAR FUNCTION
The table shows the percent (in decimal form) of battery power remaining x hours after you turn on a laptop computer.

a. Write and graph a linear function that relates y to x.
b. Interpret the slope, the x-intercept, and the y-intercept.
c. After how many hours is the battery power at75%?
Answer: a. The linear function is y = -0.2x + 1.
b. given below the explanation.
c. Battery will be 75% after 1.25 hours.

Explanation:
a. Given ,
The ordered pairs will be (0 , 1) , (2 ,0.6) , (4 , 0.2) .
The graph is
First find the slope m of the line containing the two given points (2 ,0.6) and (4 , 0.2)
m = (y₂-y₁) / (x₂-x₁)
m= (0.2 – 0.6) / (4 – 2)
m = -0.4 / 2
m = -0.2
Because the line crosses the y axis at ( 0, 1 ) , The y intercept is 1.
So , the linear function is y = -0.2x + 1.

b. Slope is -0.2 which means that as time increases by 1 hour, Battery power remaining decreases by 20% .
y intercept is 1, which means initially the battery power remaining before usage was 100%.
x intercept is 5 which means the battery remaining will be 0 after 5 hours.

c. battery percent will be 75% of 0.75 if ,
-0.2x + 1 = 0.75
0.2x = 1 – 0.75
x = 0.25/0.2
x = 1.25
Battery will be 75% after 1.25 hours.

Question 14.
MODELING REAL LIFE
The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x. The graph shows the number of calories burned by hiking.

a. Which activity burns more calories per minute?
b. You perform each activity for 45 minutes. How many total calories do you burn? Justify your answer.
Answer: a. hiking burns more calories than kayaking .
b. In kayaking, 202.5 calories are burnt per minute. and In hiking , 225 calories are burnt per minute.

Explanation:
a. The number y of calories burned after x minutes of kayaking is represented by the linear function y = 4.5x.
So, The ordered pairs of the graph are (0 , 0) , (1 , 4.5) , (2 , 9) , (3, 13.5)
Here , In kayaking burns 4.5 calories per minute .
For hiking ,
The ordered pairs of the graph are (0 , 0) , (1 , 5) , (2 , 10) , (3, 15)
Here , In hiking burns 5 calories per minute.
Thus , hiking burns more calories than kayaking .

b. Given , perform each activity for 45 minutes.
Liner function of the kayaking is y = 4.5x
substitute x = 45 in equation
y = 4.5 (45)
y = 202.5
In kayaking, 202.5 calories are burnt per minute.
Linear function of the hiking is y = 5x
substitute x = 45 in equation
y = 5 (45)
y = 225
In hiking , 225 calories are burnt per minute.

Question 15.
DIG DEEPER!
You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50. The table shows your distance at various times throughout the race. For what distances will you win the race? Explain.

Answer: you will win the race for distances greater than 190 feet

Explanation:
The distance y (in feet) your friend runs after x seconds is represented by the linear function y = 14x + 50.
The slope of the line is 14 so , your friend runs at the rate of 14 ft per second
To find your rate , the ordered pairs are (2 , 38) , (4 , 76) , (6 , 114) , (8 , 152)
First find the slope m of the line containing the two given points (2 ,38) and (4 , 76)
m = (y₂-y₁) / (x₂-x₁)
m= (76 – 38) / (4 – 2)
m = 38 / 2
m = 19
You are running at the rate of 19 ft per second.
To get the linear equation , substitute the slope in the (2 , 38) to get point slope to form a line.
Then we have , y = 19x
Now if x = 10 , to run faster then ,
y = 19(10)
y = 190 .
Your friend linear equation is y = 14x + 50 .
if x = 10 ,then
y = 14(10) + 50
y = 140 + 50
y = 190.
So , for x > 10 , means you will run farther than your friend which means you would win the race .
Therefore, you will win the race for distances greater than 190 feet.

Question 16.
REASONING
You and your friend are saving money to buy bicycles that cost $175 each. You have $45 to start and save an additional $5 each week. The graph shows the amount y(in dollars) that your friend has after x weeks. Who can buy a bicycle first? Justify your answer.

Answer:  your friend will but the bicycle first.

Explanation:
Given , your friend savings are
the ordered pairs are (0,15) and (3,39)
First find the slope m of the line containing the two given points (0,15) and (3,39)
m = (y₂-y₁) / (x₂-x₁)
m= (39 – 15) / (3 – 0)
m = 24 / 3
m = 8
Because the line crosses the y axis at ( 0, 15 ) , The y intercept is 15.
So , the linear function is y = 8x + 15.
to buy bicycles that cost $175 each
if y = 175 , then
175 = 8x + 15
8x = 175 – 15
x = 160/8
x = 20
So, your friend need 20 weeks to buy the bicycle
Given, You have $45 to start and save an additional $5 each week
So , the linear function will be y = 5x + 45.
to buy bicycles that cost $175 each
if y = 175 , then
175 = 5x + 45
5x = 175 – 45
x = 130/5
x = 26
So, you need 26 weeks to buy the bicycle.
Hence, your friend will but the bicycle first.

Question 17.
CRITICAL THINKING
Is every linear equation a linear function? Explain your reasoning.
Answer: All linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).

Question 18.
PROBLEM SOLVING
The heat index is calculated using the relative humidity and the temperature. For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F. On a summer day, the relative humidity is 75%, the temperature is 94°F, and the heat index is 124°F. Estimate the heat index when the relative humidity is 75% and the temperature is 100°F. Use a function to justify your answer.

Answer:  Heat index is 148°F

Explanation:
The form of linear equation is y = mx + c
and the slope of the function is given by m = (y₂-y₁) / (x₂-x₁)
Let y be the heat index and x be the temperature
Given , (94, 124)
For every 1 degree increase in the temperature from 94°F to 97°F at 75% relative humidity, the heat index rises 4°F
that is m = 4
Since the line passes through (94, 124) means
124 = 4x + c
124 = 4(94) + c
124 = 376 + c
c = 124 – 376
c = -252
Linear function for the heat index is y = 4x – 252
put x = 100
So, y = 4(100) – 252
y = 400 – 252
y = 148.
Finally, Heat index is 148°F.

Lesson 7.4 Comparing Linear and Non Linear Functions

EXPLORATION 1

Comparing Functions
Work with a partner. Each equation represents the height h (in feet) of a falling object after t seconds.

• Graph each equation. Explain your method.
• Decide whether each graph represents a or function.
• Compare the falling objects.

Answer: Explained below

Explanation:
a. Given, h = 300 – 15t can be written as y = 300 – 15x
h = 300 – 15t  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =300 – 15(0) = 300 . co-ordinates are (0 , 300)
if x = 1 , then y = 300 – 15(1) = 285 . co-ordinates are (1 , 285)
if x = 2 , then y = 300 – 15(2) = 270  , co-ordinates are (2 , 270)
if x = 3 , then y = 300 – 15(3) = 255  , co-ordinates are (3 , 255)
The co-ordinates (0 , 300) , (1 , 285) , (2 , 270) , (3 , 255) form a straight line .

The graph is

Given , h = 300- 16t²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =300- 16(0)²  = 300 . co-ordinates are (0 , 300)
if x = 1 , then y =300- 16(1)² = 284 . co-ordinates are (1 , 284)
if x = 2 , then y = 300- 16(2)² = 236 , co-ordinates are (2 , 236)
if x = 3 , then y = 300- 16(3)² = 252  , co-ordinates are (3 , 252)
The co-ordinates (0 , 300) , (1 , 284) , (2 , 236) , (3 , 252) does not form a straight line .

The graph is

b. For, h = 300 – 15t , The graph is linear so the so it is a function,
For h = 300- 16t² , The graph is linear so the so it is a function.

c. Sky diver has the slow fall while compared to the bowling ball , because parachute can be controlled with the wind and can be divert the destination point, and bowling ball cannot be controlled while falling.

Try It

Does the table represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = 2x – 12 is linear function.

Explanation:
Ordered pairs are (2 , -8) , (4 , -4) , (6 , 0) , (8 , 4)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (6 ,0) and (8, 4)
m = (y₂-y₁) / (x₂-x₁)
m= (4 – 0) / (8– 6)
m = 4/2
m = 2
substitute the slope in the (8 , 4) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4 = 2 ( x – 8)
y – 4 = 2x – 16
y = 2x – 16 + 4
y = 2x – 12
So ,  y = 2x – 12 is linear function.

Question 2.

Answer: y = –\(\frac{5}{3}\)x + 25 is linear function.

Explanation:
Ordered pairs are (0 , 25) , (3 , 20) , (7 , 15) , (12 , 10)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (0 ,25) and (3, 20)
m = (y₂-y₁) / (x₂-x₁)
m= (20 – 25) / (3– 0)
m = -5/3
Because the line crosses the y axis at ( 0, 25 ) , The y intercept is 25.
So , the linear function is y = –\(\frac{5}{3}\)x + 25.
So , y = –\(\frac{5}{3}\)x + 25 is linear function.

Does the equation represent a linear or nonlinear function? Explain.
Question 3.
y = x + 5
Answer: y = x + 5 is a linear function

Explanation:
Given , y = x + 5  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 5 = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 1 + 5 = 6 . co-ordinates are (1 , 6)
if x = 2 , then y = 2 + 5 = 7 , co-ordinates are (2 , 7)
The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = x + 5 is a linear function.

Question 4.
y = \(\frac{4x}{3}\)
Answer: y = \(\frac{4x}{3}\) is a linear function.

Explanation:
Given , y = \(\frac{4x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{4(0)}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = \(\frac{4(1)}{3}\) = \(\frac{4}{3}\)  = 1.3. co-ordinates are (1 , 1.3)
if x = 2 , then y = \(\frac{4(2)}{3}\) = \(\frac{8}{3}\) = 2.6 , co-ordinates are (2 , 2.6)
The co-ordinates (0 , 0) , (1 ,1.3 ) , (2 , 2.6) form a straight line .
Each x input has only one y output so it is a function .
And it forms a straight line when graphed .
So, y = \(\frac{4x}{3}\) is a linear function.

Question 5.
y = 1 – x²
Answer: y = 1 – x² is not a linear function.

Explanation:
Given , y = 1 – x²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 1 – 0² = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 1² = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 1 – 2² = -3 , co-ordinates are (2 , -3)
The co-ordinates (0 , 5) , (1 , 6) , (2 , 7) form a straight line .
Each x input has only one y output so it is a function .
And it does not forms a straight line when graphed .
So, y = 1 – x² is not a linear function.

Does the graph represent a linear or nonlinear function? Explain.
Question 6.

Answer: The graph represents a nonlinear function.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 2) , (-1 , 0) ,  (-2 , -2 ) , (-3 , -4), (0 , 1 ) , (2 , -2) , ( 3, -4 )
The inputs have more than one output ,
And points form a straight line
So , the graph is non linear function

Question 7.

Answer: The graph is a linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 0) , (-1 , -1) ,  (-2 , -2 ) , (-3 , -3), (1 , 1 ) , (2 , 2) , ( 3, 3 )
The inputs have exactly one output ,
And points form a straight line
So , the graph is a linear function.

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

IDENTIFYING FUNCTIONS Does the table or graph represent a linear or nonlinear function? Explain.
Question 8.

Answer: It is not a linear function

Explanation:
Ordered pairs are (3 , 0) , (-1 , 2) , (-5 , 4) , (-9 , 6)
Each input has exactly one output
and it does not form a straight line when graphed
So, it is not a linear function .

Question 9.

Answer: The graph is non linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , -1) , (-1 , 0) ,  (-2 , 3 ) , (1 , 0 ) , (2 , 3) .
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 10.
WHICH ONE DOESN’T BELONG?
Which equation does not belong with the other three? Explain your reasoning.

Answer: 5xy = -2 does not belong with the other three.

Explanation:
15y = 6x , y = \(\frac{2}{5}\)x , 10y = 4x .
These are evaluated as 5y = 2x
5xy = -2 , is different from 5y = 2x.

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 11.
The loudness of sound is measured in (dB). The graph shows the loudness y of a sound (in decibels) x meters from the source of the sound. Is the relationship between loudness and distance linear or nonlinear? Approximate the loudness of the sound 12 meters from the source.

Answer: The relationship between loudness and distance  is nonlinear Function. And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph.

Explanation:
As shown in the graph , the plot of the points does not form a straight line ,
Its a parabolic decay , The amount of loudness decreases with the increase in distance,
So, The relationship between loudness and distance  is nonlinear Function.

And the loudness of the sound 12 meters from the source is approximately 85dB as shown in the graph.

Question 12.
A video blogger is someone who records a video diary. A new website currently hosts 90 video bloggers and projects a gain of 10 video bloggers per month. The table below shows the actual numbers of video bloggers. How does the projection differ from the actual change?

Answer: Projections are more than the actual values

Explanation:

So, Projections are more than the actual values

Comparing Linear and Non Linear Functions Homework & Practice 7.4

Review & Refresh

Write a linear function that relates y to x.
Question 1.

Answer: The linear function is y = x – 2

In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , -2) , (1 , -1 ) , (-1 , -3) , ( 2, 0), (3 , 1) , (4 , 2) , ( 5, 3)
First find the slope m of the line containing the two given points (2 ,0) and (3, 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 0) / (3 – 2)
m = 1 .
Because the line crosses the y axis at ( 0, -2 ) , The y intercept is -2.
So , the linear function is y = x – 2 .

Question 2.

Answer: The linear function is y =\(\frac{-1}{1.5}\)x + 5.

Explanation:
Ordered pairs are (0 , 5) , (1.5 , 4) , (3 , 3) , (4.5 , 2)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1.5 ,4) and (3, 3)
m = (y₂-y₁) / (x₂-x₁)
m= (3 – 4) / (3 – 1.5)
m = -1 /1.5
Because the line crosses the y axis at ( 0, 5 ) , The y intercept is 5.
So , the linear function is y =\(\frac{-1}{1.5}\)x + 5.

The vertices of a figure are given. Draw the figure and its image after a dilation with the given scale factor. Identify the type of dilation.
Question 3.
A (- 3, 1), B (- 1, 3), C (- 1, 1); k = 3
Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 3, 1),  (- 1, 3),  (- 1, 1) these pairs form a right angle triangle
K = 3 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 3, 1) × 3 = ( -9 , 3)
(- 1, 3) × 3 = ( -3 , 9)
(- 1, 1) × 3 = (-3 , 3)
From these new ordered pairs we form a new  right angle triangle
The figure is
The New right angle triangle is larger than the original one So , its a increase .

Question 4.
J (2, 4), K (6, 10), L (8, 10), M (8, 4); k = \(\frac{1}{4}\)
Answer: It is a reduction

Explanation:
Given , (2, 4),  (6, 10),  (8, 10) ,(8,4) these pairs forms a figure
K = 0.25 , For the dilation figure multiply the 3 with the given ordered pairs , then
(2, 4) × 0.25 = (0.5, 1)
(6, 10) × 0.25 = (1.5, 2.5)
(8, 10) × 0.25 = (2, 2.5)
(8, 4) × 0.25 = (2 , 1)
From these new ordered pairs we form a new figure
The figure is
The New figure is smaller than the original , So, It is a reduction .

Concepts, Skills, & Problem Solving

COMPARING FUNCTIONS Graph each equation. Decide whether each graph represents a linear or nonlinear function. (See Exploration 1, p. 295.)
Question 5.
h = 5 + 6t Equation 1
h = 5 + 6t² Equation 2
Answer: h = 5 + 6t Equation 1 is a linear function
h = 5 + 6t² Equation 2 is a non linear function .

Explanation:
Given , h = 5 + 6t  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 5 + 6(0) = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 5 + 6(1) = 11 . co-ordinates are (1 , 11)
if x = 2 , then y = 5 + 6(2) = 17 , co-ordinates are (2 , 17)
if x = 3 , then y = 5 + 6(3) = 23 , co-ordinates are (3 , 23)
The co-ordinates (0 , 5) , (1 , 11) , (2 , 17) , (3 , 23) form a straight line .

Given , h = 5 + 6t²  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 5 + 6(0)² = 5 . co-ordinates are (0 , 5)
if x = 1 , then y = 5 + 6(1)² = 11 . co-ordinates are (1 , 11)
if x = 2 , then y = 5 + 6(2)² = 26 , co-ordinates are (2 , 26)
if x = 3 , then y = 5 + 6(3)² = 59 , co-ordinates are (3 , 59)
The co-ordinates (0 , 5) , (1 , 11) , (2 , 26) , (3 , 59) does not form a straight line .

The graph of both equations is
So, h = 5 + 6t Equation 1 is a linear function
h = 5 + 6t² Equation 2 is a non linear function .

Question 6.
y = – \(\frac{x}{3}\) Equation 1
y = – \(\frac{3}{x}\) Equation 2
Answer:  y = – \(\frac{x}{3}\) Equation 1 is a linear function
y = – \(\frac{3}{x}\) Equation 2 is a non linear function.

Given , y =- \(\frac{x}{3}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- \(\frac{0}{3}\) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – \(\frac{1}{3}\) = – 0.3 . co-ordinates are (1 , – 0.3 )
if x = 2 , then y = – \(\frac{2}{3}\) = – 0.6 , co-ordinates are (2 ,-0.6)
if x = 3 , then y = – \(\frac{3}{3}\) = -1 , co-ordinates are (3 , -1)
The co-ordinates (0 , 0) , (1 , -0.3) , (2 , -0.6) , (3 , -1) form a straight line .

Given , y =- \(\frac{3}{x}\) , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- \(\frac{3}{0}\) = no number
if x = 1 , then y = – \(\frac{3}{1}\) = – 3 . co-ordinates are (1 , – 1 )
if x = 2 , then y = – \(\frac{3}{2}\) = – 1.5 , co-ordinates are (2 ,-1.5)
if x = 3 , then y = – \(\frac{3}{3}\) = -1 , co-ordinates are (3 , -1)
The co-ordinates (1 , -1) , (2 , -1.5) , (3 , -1) form a straight line .

The graph of both the equations is
So,  y = – \(\frac{x}{3}\) Equation 1 is a linear function
y = – \(\frac{3}{x}\) Equation 2 is a non linear function.

IDENTIFYING FUNCTIONS FROM TABLES Does the table represent a linear or nonlinear function? Explain.
Question 7.

Answer:  linear function is y = 4x + 4.

Explanation:
Ordered pairs are (0 , 4) , (1 , 8) , (2 , 12) , (3 , 16)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (2 , 12) and (3 , 16)
m = (y₂-y₁) / (x₂-x₁)
m= (16 – 12) / (3– 2)
m = 4/1
m = 4
Because the line crosses the y axis at ( 0, 4 ) , The y intercept is 4.
So , the linear equation is y = 4x + 4.
And it is a linear function.

The graph is

Question 8.

Answer: y = 4x – 6 is linear function.

Explanation:
Ordered pairs are (6 , 21) , (5 , 15) , (4 , 10) , (3 , 6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (4 , 10) and (3 , 6)
m = (y₂-y₁) / (x₂-x₁)
m= (6 – 10) / (3– 4)
m = -4/-1
m = 4
substitute the slope in the(4 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = 4 ( x – 4)
y – 10 = 4x – 16
y = 4x – 16 + 10
y = 4x – 6
So ,  y = 4x – 6 is linear function.

The graph is

IDENTIFYING FUNCTIONS FROM EQUATIONS Does the equation represent a linear or nonlinear function? Explain.
Question 9.
2x + 3y = 7
Answer: The function is linear when m = \(\frac{-2}{3}\) and c = \(\frac{7}{3}\)

Explanation:
Given ,2x + 3y = 7
3y = 7 – 2x
y = \(\frac{-2}{3}\)x+ \(\frac{7}{3}\)
So, The function is linear when m = \(\frac{-2}{3}\) and c = \(\frac{7}{3}\)

Question 10.
y + x = 4x + 5
Answer: The function is linear when m = 3 and c = 5 .

Explanation:
Given , y + x = 4x + 5
y = 4x – x + 5
y = 3x + 5
So, The function is linear when m = 3 and c = 5 .

Question 11.
y = \(\frac{8}{x^{2}}\)
Answer: The function is linear when m = 8 and c = 0 .

Explanation:
Given , y = \(\frac{8}{x^{2}}\)
slope m = 8
c = 0
So, The function is linear when m = 8 and c = 0 .

IDENTIFYING FUNCTIONS FROM GRAPHS Does the graph represent a linear or nonlinear function? Explain.
Question 12.

Answer: The graph is linear function

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 1) , (2 , 0) ,  (4 , -1 ) , (-2 , 2), ( -4, 3 )
The inputs have exactly one output ,
And points form a straight line
So , the graph is linear function

Question 13.

Answer: The graph is non linear function.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 0) , (-1 , -1) ,  (-4 , -2 ) , (1 , 1), ( 4, 2 )
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 14.
IDENTIFYING A FUNCTION
The graph shows the volume V (in cubic feet) of a cube with an edge length of x feet. Does linear nonlinear the graph represent a linear or nonlinear function? Explain.

Answer: The graph is non linear function

n order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (1 , 1) , (2 , 8) ,  (3 , 27 ) , (4 , 64)
The inputs have exactly one output ,
And points does not form a straight line
So , the graph is non linear function

Question 15.
MODELING REAL LIFE
The frequency y (in terahertz) of a light wave is a function of its wavelength x (in nanometers). Is the function relating the wavelength of light to its frequency linear or nonlinear?

Answer: The function is a non linear function

Explanation:
table is as follows
change in x is constant but change in y is not constant , it is increasing
So, the function is a non linear function .

Question 16.
DIG DEEPER!
The table shows the cost (in dollars) of pounds of sun flower seeds.

a. What is the missing -value that makes the table represent a linear function?
b. Write a linear function that represents the cost of x pounds of seeds. Interpret the slope.
c. Does the function have a maximum value? Explain your reasoning.
Answer:  a. 3 pounds = $4.2
b. y = 1.4x  is linear function.
c.  If y has maximum value then the x also has maximum value.

Explanation:
a. As per the table 1 pound = $1.4
2 pounds = $2.8
3pounds = $4.2
4 pounds = $5.6
So, the price is increasing with weight of the seeds.

b. Ordered pairs are (2 , 2.8) , (3 , 4.2) , (4 , 5.6)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 4.2) and (4 , 5.6)
m = (y₂-y₁) / (x₂-x₁)
m= (5.6 – 4.2) / (4 – 3)
m = 1.4
substitute the slope in the (3 , 4.2) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 4.2 = 1.4 ( x – 3)
y – 4.2 = 1.4x – 4.2
y = 1.4x – 4.2 + 4.2
y = 1.4x
So ,  y = 1.4x  is linear function.

c. As shown in the table , and the function if y increases then x also increases with respect to the y
So, if y has maximum value then the x also has maximum value.

Question 17.
MODELING REAL LIFE
A birch tree is 9 feet tall and grows at a rate of 2 feet per year. The table shows the height h (in feet) of a willow tree after x years.

a. Does the table represent a linear or nonlinear function? Explain.
b. Which tree is taller after 10 years? Explain.
Answer: There is no linear relationship between x and y .

Explanation:
Table is as follows
Change in y is constant but change in x is increasing , not a constant
Hence, there is no linear relationship between x and y .

Question 18.
CRITICAL THINKING
In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
a. Determine whether the function relating the year to the number of viewers is linear or nonlinear for each show.
b. Which show has more viewers in its sixth year?
Answer: a. The function relating the year to the number of viewers is linear
b. Both shows  has same number of viewers in the sixth year .

Explanation:
a. Given, In their first year, Show A has 7 million viewers and Show B has 5 million viewers. Each year, Show A has 90% of the viewers it had in the previous year. Show B loses 200,000 viewers each year.
For show A
So , In first year = 7
2 year = 90% of 7 = 6.3
3 year = 90% of 6.3 = 5.6
4 year = 90% of 5.6 = 5.04
5 year = 90% of 5 = 4.5
6 year = 90% of 4.5 = 4.05
So the ordered pairs are (1 , 7) , (2 , 6.3) , (3 , 5.6) , (4 , 5), (5 , 4.5) , (6 , 4)

For show B
In first year = 5 , As the viewers reduces by 2,00,000 in 5M
2 year = 5 – 0.2 = 4.8
3 year = 4.8 – 0.2 = 4.6
4 year = 4.6 – 0.2 = 4.4
5 year = 4.4 – 0.2 = 4.2
6 year = 54.2 – 0.2 = 4
So the ordered pairs are (1 , 5) , (2 , 4.8) , (3 , 4.6) , (4 , 4.4), (5 , 4.2) , (6 , 4)
As the year increases the viewers are also decreasing constantly as per the individual shows
So, The function relating the year to the number of viewers is linear .

b. As shown in part a , the ordered pairs having (6,4) represents the number of viewers to the year
So, Both shows  has same number of viewers in the sixth year .

Question 19.
NUMBER SENSE
The ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. Graph the ordered pairs and describe the pattern. Is the function linear or nonlinear?
b. Write an equation that represents the function.
Answer: a. The graph is shown below and function is linear
b. The linear equation is y = 7x – 1.

Explanation:
Given, ordered pairs represent a function. (0,- 1), (1, 0), (2, 3), (3, 8), and (4, 15)
a. the graph is 
Each input has exactly one output and it forms a straight line So, the graph is linear
b. First find the slope m of the line containing the two given points (3 ,8) and (4, 15)
m = (y₂-y₁) / (x₂-x₁)
m= (15 – 8) / (4– 3)
m = 7/1
m = 7
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear equation is y = 7x – 1.

Lesson 7.5 Analyzing and Sketching Graphs

EXPLORATION 1

Matching Situations to Graphs
Work with a partner. Each graph shows your speed during a bike ride. Match each situation with its graph. Explain your reasoning.

a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. Analyze Relationships
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed.
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Answer: a – C ,
b – A ,
c – D ,
d – B ,

Explanation:
a. You increase your speed, then ride at a constant speed along a bike path. You then slow down until you reach your friend’s house. The graph C has the perfect graph representing the situation of given question.
b. You increase your speed, then go down a hill. You then quickly come to a stop at an intersection.
Because The graph A has the bike speed representing the situation for the time .
c. You increase your speed, then stop at a store for a couple of minutes. You then continue to ride, increasing your speed. Thus, The graph D is the final answer for the question
d. You ride at a constant speed, then go up a hill. Once on top of the hill, you increase your speed.
Because of the speed with respect to time the graph B is the correct answer for the question.

EXPLORATION 2

Interpreting a Graph
Work with a partner. Write a short paragraph that describe show the height changes over time in the graph shown. What situation can this graph represent?

Answer: The Graph can be representing a situation for low and high tides of the Ocean

Explanation:
As shown in the figure, The graph is plotted between the height and time,
We can take an example of an Ocean for its waves , As the time passes at the morning of a normal day, The waves of the ocean start rising higher at a period of time, and for the time being maintaining a peak height then drops to a lower height at a particular intervals of time , this process takes place for a while and vise versa.
Thus, the Graph can be representing a situation for low and high tides of the Ocean

Try It

Question 1.
The graph shows the location of a pelican relative to your location.

a. Describe the path of the pelican.
b. Write an explanation for the decrease in the vertical distance of the pelican.
Answer: Both of them are explained below.

Explanation:
a. The path of the pelican is flying in the air , As they always fly in line and the amazing thing is the deeper the prey the higher they dive.
The graph shows the relationship between the horizontal distance that is the height from the land, vertical distance is the point from where its destination point is located, so at the starting point of the flight it has more distance from the ground means flying at a higher level , as the time passes it reaches to the closer point of its destination point so the altitude of the flight decreases with the decrease in the vertical distance and at a particular distance reaches its point of destination.

b. The decrease in the vertical distance of the pelican. is due to its flight to the destination point as it requires to stop flying to reach it, so in order to have a smooth landing on the ground , the bird gradually decreases its speed by decreasing its altitude.

Question 2.
A fully-charged battery loses its charge at a constant rate until it has no charge left. You plug it in, and it fully recharges at a constant rate. Then it loses its charge at a constant rate until it has no charge left. Sketch a graph that represents this situation.
Answer:  The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the battery charge ,
A fully-charged battery loses its charge at a constant rate until it has no charge left. So, line segment starts from 100 and decreases until it touches the x-axis.
You plug it in, and it fully recharges at a constant rate. Thus, line segment increases at a constant rate until it reaches 100
Then it loses its charge at a constant rate until it has no charge left. line segment decreases again at a constant rate until it again touches the x-axis .

Self-Assessment for Concepts & Skills
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.
ANALYZING GRAPHS
The graph shows the growth rate of a plant over time.

a. Describe the change in growth rate.
b. Write an explanation for the decrease in growth rate and the increase in growth rate.
Answer: the answers are given below

Explanation:
a. the change in growth rate of a plant over the time is given by its size and height , So as the time passes the growth rate is constant from the the start and from a particular time the growth rate has been dropping slightly due to external or internal reasons of a plant and again at some time the growth rate is increasing at a constant rate until it reaches to its perfect growth of a plant.

b. The decrease in growth rate of the plant is due to some external causes like weather, rain, sunlight , watering, and the soil may effect its growth rate and the increase in growth rate is probably due to its soil fertility and sufficient sunlight providing sufficient chlorophyll.

Question 4.
SKETCHING GRAPHS
As you snowboard down a hill, you gain speed at a constant rate. You come to a steep section of the hill and gain speed at a greater constant rate. You then slow down at a constant rate until you come to a stop. Sketch a graph that represents this situation.
Answer: The graph is

Explanation:
In the graph , let the x-axis be time and y-axis be the speed ,
As you snowboard down a hill, you gain speed at a constant rate, line segment decreases at a constant rate
You come to a steep section of the hill and gain speed at a greater constant rate, line segment becomes steeper i.e., the line segment decreases at a high constant rate.
You then slow down at a constant rate until you come to a stop, line segment becomes flatter i.e., the constant rate of decrease becomes less until it touches its x-axis

Self-Assessment for Problem Solving
Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.
Two rowing teams are in a race. The graph shows their distances from the finish line over time. Describe the speed of each team throughout the race. Then determine which team finishes first.

Answer: Team B will finishes race first.

Explanation:
Team A , The relationship between the time and distance from the finish line is given in the graph,
At starting point Team A has maintained a fair speed at the Beginning of the race and has been a little slow while reaching out to the destination point, and for a while they have been balancing the speed with the distance representing a curving point in the graph and directly dropping to the finish line drastically creating a slope, until it reaches in the x-axis line.
Team B , The relationship between the time and distance from the finish line is given in the graph,
As same as the Team A , Team B has a perfect start but it has been a way different them Team A because Team B has a game plan to win the race, as shown in the graph they have maintained a constant speed while reaching out to the destination and also having a smooth drift at a level of decreasing their distance from the finish line.

Team B will  finishes the race first because they are having a constant and smooth decreasing speed which comes to an end gradually at the finishing line.

Question 6.
DIG DEEPER!
The graphs show the movements of two airplanes over time. Describe the movement of each airplane.

Answer: Detailed explanation is given below.

Explanation:
As shown in the graph , x-axis is time and y-axis be the height above ground
Airplane A, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.

Airplane B, the line segment at a constant rate at the time of starting of the takeoff and then drops to a point while decreasing in the height to the ground for landing and for a constant time it is at the ground level until again it takes off  having an increase in the height from the ground level , at last it maintains a constant speed.
It is as same as the airplane A.

Analyzing and Sketching Graphs Homework & Practice 7.5

Review & Refresh

Does the table or equation represent a linear or nonlinear function? Explain.
Question 1.

Answer: y = -0.5x + 11.5 is a linear function.

Explanation:
Ordered pairs are (-5 , 14) , (-1 , 12) , (3 , 10) , (7 , 8)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 10) and (7 , 8)
m = (y₂-y₁) / (x₂-x₁)
m= (8 – 10) / (7 – 3)
m = -2/4
m = -0.5
substitute the slope in the(3 , 10) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 10 = -0.5 ( x – 3)
y – 10 = -0.5x + 1.5
y = -0.5x + 1.5 + 10
y = -0.5x + 11.5
So , the linear equation is y = -0.5x + 11.5
And it is a linear function.

The graph is

Question 2.
y = x² + 8
Answer: The function is linear  when m= 1 and c = 8.

Explanation:
Given , y = x² + 8 ,
slope m = 1
c = 8
So, the function is linear when m = 1 and c= 8.

Graph the linear equation.
Question 3.
– 4x + y = – 1
Answer: The graph is

Explanation:
we can write – 4x + y = – 1 as y = 4x – 1
Given , y = 4x – 1  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 4(0) – 1 = -1 . co-ordinates are (0 , -1)
if x = 1 , then y = 4(1) – 1 = 3 . co-ordinates are (1 , 3)
if x = 2 , then y = 4(2) – 1 = 7 , co-ordinates are (2 , 7)
if x = 3 , then y = 4(3) – 1= 11 , co-ordinates are (3 , 11)
The co-ordinates (0 , -1) , (1 , 3) , (2 , 7) , (3 , 11) form a straight line .

Question 4.
2x – 3y = 12
Answer: The graph is

Explanation:
we can write  2x – 3y = 12 as y = \(\frac{2x-12}{3}\) or y = \(\frac{2}{3}\)x – 4
Given , y =\(\frac{2}{3}\)x – 4 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = \(\frac{2}{3}\)0 – 4= – 4 . co-ordinates are (0 , -4)
if x = 1 , then y = \(\frac{2}{3}\)1 – 4 = 0.66 – 4 = -3.3 . co-ordinates are (1 , -3.3)
if x = 2 , then y = \(\frac{2}{3}\)2 – 4 =0.66(2) – 4 =1.3 – 4 = -2.6, co-ordinates are (2 , -2.6)
if x = 3 , then y = \(\frac{2}{3}\)3 – 4 = 0.66(3) – 4 = 1.98 – 4 = -2.0  , co-ordinates are (3 , -2.0)
The co-ordinates (0 , -4) , (1 , -3.3) , (2 , -2.6) , (3 , -2) form a straight line .

Question 5.
5x + 10y = 30
Answer: The graph is

Explanation:
5x + 10y = 30 can be written as y = -0.5x + 3
take 5 common on both sides we get
x + 2y = 6
y = \(\frac{-x + 6}{2}\)
y = \(\frac{-x}{2}\) + 6
y = -0.5x + 3
Given , y =-0.5x + 3 , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = -0.5(0) + 3 = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = -0.5(1) + 3= 2.5 . co-ordinates are (1 , 2.5)
if x = 2 , then y = -0.5(2) + 3 = 4 , co-ordinates are (2 , 4)
if x = 3 , then y = -0.5(3) + 3 = 4.5  , co-ordinates are (3 , 4.5)
The co-ordinates (0 , 3) , (1 , 2.5) , (2 , 4) , (3 , 4.5)does not form a straight line .

Concepts, Skills, &Problem Solving

MATCHING DESCRIPTIONS WITH GRAPHS The graph shows your speed during a run. Match the verbal description with the part of the graph it describes. (See Exploration 1, p. 301.)

Question 6.
You run at a constant speed.
Answer: C

Explanation:
Because the line segment of the graph at point C show that the running speed is constant for a particular time ,
Thus forming a straight horizontal line.

Question 7.
You slow down at a constant rate.
Answer: D

Explanation:
Because the line segment of the graph at point D show that the running speed is decreasing at a constant rate for a particular time ,
Thus forming a straight steep line down the time axis.

Question 8.
You increase your speed at a constant rate.
Answer: A

Explanation:
Because the line segment of the graph at point A show that the running speed is increasing at a constant rate at a starting point of the race on time ,
Thus forming a slope in the graph.

Question 9.
You increase your speed at a faster and faster rate.
Answer: B

Explanation:
Because the line segment of the graph at point B show that the running speed is increasing at a faster rate after starting the race and maintaining a gradual growth of the speed and after reaching the next point speed is doubled from before ,
Thus forming a slope with a curve in the graph.

ANALYZING GRAPHS Describe the relationship between the two quantities.
Question 10.

Answer: As the Time passes there will be increase in the volume.

Explanation:
The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

So, As the Time passes there will be increase in the volume.

Question 11.

Answer: As the times passes Dollars are maintaining imbalance.

Explanation:
The relationship between the time and dollars is given in the graph, As we all know money is never ever constant with time , As if it only increases or decreases or having both simultaneously , in this graph the line segment is having a steep and at some point of time it is maintaining a slight growth constantly with the time.

So, As the times passes Dollars are maintaining imbalance.

Question 12.

Answer: An engine power is directly proportional to the engine speed and its horse power

Explanation:
The relationship between the engine speed and horse power is given in the graph, Generally every automobile is is defined as the best for its horse power which is the heart of the engine and it highlights the speed of the vehicle, Here engine power is defined by the horse power and the engine speed the line segment is having a curve increment in the horse power due to the increase in engine speed.

So, An engine power is directly proportional to the engine speed and its horse power

Question 13.

Answer: As time increases the process of grams decaying will be faster.

Explanation:
The relationship between grams and time is given in the graph, its obvious that every product has its own expiry date, and if it crosses that its starts to decay, the graph implies that with the increase time the quality of the gram decreases or grams start to decay . The line segment in the graph shows that the gradually decrease indicating the spoiling rate of the grams  with rate of change of time.

So, As time increases the process of grams decaying will be faster.

Question 14.

Answer: At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Explanation:
The graph shows the relationship between the length of the hair and time taken to the growth of the hair, of course hair growth is not constant every time, here we have the graph with the line segment  not constant and having breaks at the times of interval.

So, At the particular intervals of time hair growth has stopped and at regular intervals again starts growing with respect to the time.

Question 15.

Answer: In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Explanation:
The relationship between the balance of the loan with the time period of the loan to be cleared, The loan should be cleared in the time limit and should maintain a neat balance, every increase in time period the balance is debited from the loan , there will be decrease in the balance and gaps are occurred in the graph.

so, In a period of time the balance will be reduced gradually at regular intervals maintaining a constant balance to clear the loan.

Question 16.
ANALYZING GRAPHS
Write an explanation for the relationship shown in the graph in Exercise 10.
Answer: The graph shows the relation between the volume and time of a Balloon , To fill up the balloon with air, with the inlet of air increases in volume and the time taken to fill the balloon with air is shown ,
The line segment starts at initial point stating the balloon at a no air state, then gradually increases with constant halts having constant volume at a particular time and vice versa.

Question 17.
MODELING REAL LIFE
The graph shows the natural gas usage for a house.

a. Describe the change in usage from January to March.
b. Describe the change in usage from March to May.
Answer: a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.
b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

Explanation:
a. The change in usage from January to March, As shown in the graph the usage is at peaks in the month of January and started to decrease after that month and continuing to decrease in the month of March.

b.  The change in usage from March to May, As shown in the graph the usage is continuing to decrease in the month of March and started to increase again in the month of May and maintaining to use highly from the month of May.

SKETCHING GRAPHS Sketch a graph that represents the situation.
Question 18.
The value of a television decreases at a constant rate, and then remains constant.
Answer: The graph is

Explanation:
Draw the axis and label the x- axis as time and y- axis as value, then sketch the graph,
The value of the television decreases at a constant rate: line segment starts to decrease at a constant rate,
And then remains constant, after reaching a certain value : line segment becomes parallel to horizontal axis.

Question 19.
The distance from the ground changes as your friend swings on a swing.
Answer: The graph is

Explanation:
Your friend starts close to the ground and then swings up. Then she falls back down close to the ground again and swings back . When she swings back, she gets higher than when she was swinging forward, she then starts to swing forward again getting close to the ground and then going up even higher than when she was swinging backward, she continues to getting higher and higher every time she swings forwards and backwards,

Question 20.
The value of a rare coin increases at a faster and faster rate.
Answer: The graph is

Explanation:
Draw the Axis and label them as x-axis as time and y – axis as distance,
The value of a rare coin increases at a faster and faster rate , so the curve moves upwards at an increasing rate.

Question 21.
You are typing at a constant rate. You pause to think about your next paragraph and then you resume typing at the same constant rate.
Answer: The graph is

Explanation:
A constant rate means that portion of the graph is linear , pausing means the number of words stays constant, typing again at the same constant rate means the last piece of the graph is linear again with the same slope as the first portion of the graph.

Question 22.
CRITICAL THINKING
The graph shows the speed of an object over time.

a. Sketch a graph that shows the distance traveled by the object over time.
b. Describe a possible situation represented by the graphs.
Answer: a. The distance and time are directly proportional to each other.
b. As time passes the speed and time are relatively balancing each other in the graph.

Explanation:
a. The graph is 
In this graph the relationship between distance and time is shown, for example , let the object be a bike, the time taken to reach the destination for the bike is directly proportional to the distance travelled , So as time passes the distance is gradually increasing from the starting point.

So, the distance and time are directly proportional to each other.

b. Th graph shown , is the relationship between the speed and the time , let the object moving be Train,
it is running between the station so it has to be halted in the stations to be listed in the stoppings , So the line segment is started with a constant speed with the time and again at the time interval dropping the speed with respect to time it has maintaining the same speed .

So, As time passes the speed and time are relatively balancing each other in the graph.

Question 23.
MODELING REAL LIFE
The graph shows the average scores of two bowlers from the start of a season to the end of the season.

a. Describe each bowler’s performance.
b. Who had a greater average score most of the season? Who had a greater average score at the end of the season?
c. Write an explanation for the change in each bowler’s average score throughout the bowling season.
Answer: All the answers are explained below

Explanation:
a.  Bowler A : As the graph represent the relationship between the score and the week, bowler A has started with the good take off and having able to grasp the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B.

Bowler B : As the graph represent the relationship between the score and the week, bowler B has started with the good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week.

b.  Bowler A and Bowler B had a greater average score most of the season, but Bowler A had a greater average score at the end of the season

c. Bowler A has the same energy his performance on the goals are increasing rapidly over the week and had the good score than Bowler B. so it has a smaller change in average’s score in the bowling season .
While Bowler B has good take off but as the time passes the performance of the  bowler is has been intended to decrease his scores gradually over the week. so he has a drastic change in average’s score in the bowling season .

Question 24.
DIG DEEPER!
You can use a supply and demand model to understand how the price of a product changes in a market. The supply curve of a particular product represents the quantity suppliers will produce at various prices. The demand curve for the product represents the quantity consumers are willing to buy at various prices.

a. Describe and interpret each curve.
b. Which part of the graph represents a surplus? Explain your reasoning.
c. The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Suppose that demand for a product suddenly increases, causing the entire demand curve to shift to the right. What happens to the equilibrium point?
Answer:  All of them are explained below .

Explanation:
a. The supply curve of a particular product represents the quantity suppliers will produce at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices increases gradually Quantity increases .
The demand curve for the product represents the quantity consumers are willing to buy at various prices, As shown in the graph, the relationship between the price and quantity is given , so if prices decreases with increase in Quantity .

b. The graph does not implies any surplus because each demand and supply is given by their respective curve over the prices and quantity

c. As shown in the graph, The curves intersect at the equilibrium point, which is where the quantity produced equals the quantity demanded. Given, that demand for a product suddenly increases, causing the entire demand curve to shift to the right. Then the equilibrium point will be pointed where the two curves meet after the change in the demand graph so change in the supply graph is also possible.

Functions Connecting Concepts

Using the Problem-Solving Plan
Question 1.
The table shows the lengths x (in inches) and weights y(in pounds) of several infants born at a hospital. Determine whether weight is a function of length. Then estimate the weight of an infant that is 20 inches long.

Understand the problem.
You know the lengths and weights of several infants. You are asked to determine whether weight is a function of length and to estimate the weight of a 20-inch-long infant.

Make a plan.
Determine whether any of the lengths are paired with more than one weight. Then use a graphing calculator to find an equation that represents the data. Evaluate the equation when x = 20 to estimate the weight of a 20-inch-long infant.

Solve and check.
Use the plan to solve the problem. Then check your solution.
Answer: Weight is the function of the length

Explanation:
From the table we have , Each length has only one weight , so weight is a function of length.
First find the slope m of the line containing the two given points (19.3 , 7.3) and (18.9 , 6.5)
m = (y₂-y₁) / (x₂-x₁)
m= (6.5 – 7.3) / (18.9 – 19.3)
m = 0.2
substitute the slope in the (19.3 , 7.3) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 7.3 = 0.2 ( x – 19.3)
y – 7.3 = 0.2x – 3.86
y = 0.2x – 3.86 + 7.3
y = 0.2x + 3.4
So ,  y = 0.2x + 3.4 is linear function.

For x = 20 ,
y = 0.2 (20) + 3.4
y = 4 + 3.4
y = 7.4

So, The weight of an infant that is 20 inches long. is 7.4.

Question 2.
Each mapping diagram represents a linear function. At what point do the graphs of the functions intersect? Justify your answer.

Answer:  The point of intersection is (-1, -4)

Explanation:
Function 1 – Ordered pairs are ( -8 , 24 ) , ( -3 , 4 ) , ( -1 , -4 ) , ( 1 , -12) .
Function 2 – Ordered pairs are ( 6 , 17 ) , ( 10 , 29 ) , ( 13 , 38 ) , ( 15 , 44 ) .
Graph the points we get, So, The point of intersection is (-1,-4).

Performance Task

Heat Index
At the beginning of this chapter, you watched a STEAM Video called “Apparent Temperature.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

Answer:

Functions Chapter Review

Review Vocabulary

Write the definition and give an example of each vocabulary term.

Input: Ordered pairs can be used to show inputs and outputs , inputs are represented by x

Output: Ordered pairs can be used to show inputs and outputs , Outputs are represented by y

Relation: A relation pairs inputs with outputs

Mapping diagram: A relation can be represented by ordered pairs or mapping diagrams.

Function: The relation that pairs each input with exactly one output is a function.

Function rule: it is an equation, that describes the relationship between inputs(independent variables) and outputs(dependent variables).

Linear function: A linear function is a function whose graph is a straight line i.e., non vertical line . A linear can be written in the form y = mx + c , where m is the slope and c is the y intercept

Non linear function: The graph of a linear function shows a constant rate of change, A non linear function does not have a constant rate of change, So its graph is a not a line.

Graphic Organizers
You can use an Example and Non-Example Chart to list examples and non-examples of a concept. Here is an Example and Non-Example Chart for functions.

Choose and complete a graphic organizer to help you study the concept.

1. linear functions
2. nonlinear functions
3. linear functions with positive slope
4. linear functions with negative slope

Answer: 1. linear functions

Chapter Self-Assessment

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

7.1 Relations and Functions (pp. 275–280)
Learning Target: Understand the concept of a function.

List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.
Question 1.

Answer: The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) and The relation is not a function .

Explanation:
As shown , The ordered pairs are ( 1 , -4 ) , ( 3 , 6 ) , ( 5 , 0 ) , ( 7 , 6 ) , ( 7 , 8 ) .
The input 7 has more than one output,
So, The relation is not a function .

Question 2.

Answer: ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ) and The relation is a function .

Explanation:
As shown , The ordered pairs are ( 0 , 0 ) , ( 1 , 10 ) , ( 2 , 5 ) , ( 3 , 15 ).
Each input has exactly one output ,
So, The relation is a function .

Question 3.

Answer: The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) and The relation is not a function

Explanation:
As shown , The ordered pairs are ( -1 , 0 ) , ( -1 , 1 ) , ( 0 , 1 ) , ( 1 , 2 ), ( 3 ,3 ) .
The input -1 has more than one output ,
So, The relation is not a function .

Question 4.
For ordered pairs that represent relations, which coordinate represents the input? the output?

Answer: x coordinate is the input and y coordinate is the output

Explanation:
Ordered pairs from the given graph are ( 2 , 7 ) , ( 3 , 7 ) , ( 4 , 5 ) , ( 5 , 5 ) , ( 6 , 3 ) .
So , x coordinate is the input and y coordinate is the output

Question 5.
Draw a mapping diagram that represents the relation shown in the graph. Then determine whether the relation is a function. Explain.
Answer:

Explanation:
The mapping diagram is
each input has more than one output
So, relation is not a function.

Question 6.
The mapping diagram represents the lengths (in centimeters) of a rubber band when different amounts of force (in Newtons) are applied.

a. Is the length of a rubber band a function of the force applied to the rubber band?
b. Describe the relationship between the length of a rubber band and the force applied to the rubber band.
Answer:  a. Yes
b. For every increase in 0.7 in input there is an increment of 2 in output.

Explanation:
a. The ordered pairs are  ( 0 , 5 ) , ( 0.7 , 7 ) , ( 1.4 , 9 ) , ( 2.1 , 11 )
Each input has exactly one output
So, the length of a rubber band a function of the force applied to the rubber band.

b. For every increase in 0.7 in input there is an increment of 2 in output.

7.2 Representations of Functions (pp. 281–288)
Learning Target: Represent functions in a variety of ways.

Write a function rule for the statement.
Question 7.
The output is two less than the input.
Answer: y = x – 2

Explanation:
Let us say x is input and y is output , then
The output is two less than the input, will be
y = x – 2

Question 8.
The output is two more than one-fourth of the input.
Answer: y = \(\frac{x}{4}\) + 2

Explanation:
Let us say x is input and y is output , then
The output is two more than one-fourth of the input, will be
y = \(\frac{x}{4}\) + 2

Find the value of y for the given value of x.
Question 9.
y = 2x – 3; x = – 4
Answer: y = -8

Explanation:
Given, y = 2x
substitute x = -4 , we get
y = 2(-4)
y = -8.

Question 10.
y = 2 – 9x ; x = \(\frac{2}{3}\)
Answer: y = – 3.4

Explanation:
Given , y = 2 – 9x
substitute x = \(\frac{2}{3}\) , we get
y = 2 – 9 (0.6)
y = 2 – 5.4
y = – 3.4

Question 11.
y = \(\frac{x}{3}\) + 5; x = 6
Answer: y = 7.

Explanation:
Given, y = \(\frac{x}{3}\) + 5
substitute x = 6 , we get
y = \(\frac{6}{3}\) + 5
y = 2 + 5
y = 7.

Graph the function.
Question 12.
y = x + 3
Answer: The graph is

Explanation:
Given , y = x + 3  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 0 + 3 = 3 . co-ordinates are (0 , 4)
if x = 1 , then y = 1 + 3  = 4 . co-ordinates are (1 , 5)
if x = 2 , then y = 2 + 3 = 5 , co-ordinates are (2 , 6)
if x = 3 , then y = 3 + 3 = 6 , co-ordinates are (3 , 7)
The co-ordinates (0 , 3) , (1 , 4) , (2 , 5) , (3 , 6) form a straight line .

Question 13.
y = – 5x
Answer: The graph is

Explanation:
Given , y = – 5x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =- 5(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = – 5(1)  = – 5 . co-ordinates are (1 , – 5)
if x = 2 , then y = – 5(2) = -10 , co-ordinates are (2 , -10)
if x = 3 , then y =- 5(3) = -15 , co-ordinates are (3 , -15)
The co-ordinates (0 , 0) , (1 , -5) , (2 , -10) , (3 , -15) form a straight line .

Question 14.
y = 3 – 3x
Answer: The graph is

Explanation:
Given , y =3 – 3x , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 3 – 3(0) = 3 . co-ordinates are (0 , 3)
if x = 1 , then y = 3 – 3(1)  = 0 . co-ordinates are (1 , 0)
if x = 2 , then y = 3 – 3(2) = – 3 , co-ordinates are (2 , – 3)
if x = 3 , then y =3 – 3(3) = – 6 , co-ordinates are (3 , – 6)
The co-ordinates (0 , 3) , (1 , 0) , (2 , – 3) , (3 , – 6) form a straight line .

Question 15.
An online music store sells songs for $0.90 each.
a. Write a function that you can use to find the cost of buying songs.
b. What is the cost of buying 5 songs?
Answer: a. C = 0.90s
b. $4.5

Explanation:
a. The total cost is equal to the cost of each song times the number of songs, if each song is $0.90,
Then the total cost C of s songs is C = 0.90s.

b. Substituting s= 5 in C = 0.90s we get,
C = 0.90(5) = 4.5.
So, cost of 5 songs is $4.5.

7.3 Linear Functions (pp. 289–294)
Learning Target: Use functions to model linear relationships.

Use the graph or table to write a linear function that relates y to x.
Question 16.

Answer: The linear function is y = \(\frac{1}{3}\)x + 3.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (3 , 4) , (0 , 3 ) , (-3 , 2) , ( -6, 1 )
First find the slope m of the line containing the two given points (0 ,3) and (-3, 2)
m = (y₂-y₁) / (x₂-x₁)
m= (2 – 3) / (-3 – 0)
m = -1 / -3 .
m = 1/3 .
Because the line crosses the y axis at ( 0, 3 ) , The y intercept is 3.
So , the linear function is y = \(\frac{1}{3}\)x + 3.

Question 17.

Answer: The linear function is y = −(0)x -7.

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-2 , -7) , (0 , -7 ) , (2 , -7) , ( 4 , -7 )
First find the slope m of the line containing the two given points (0 ,-7) and (2, -7)
m = (y₂-y₁) / (x₂-x₁)
m= (-7 – (-7)) / (2 – 0)
m = 0 .
Because the line crosses the y axis at ( 0, -7 ) , The y intercept is -7.
So , the linear function is y = −(0)x -7.

Question 18.
The table shows the age x (in weeks) of a puppy and its weight y (in pounds).

a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
c. After how many weeks will the puppy weigh 33 pounds?
Answer: a. y = \(\frac{3}{2}\)x + 3
b. 3 pounds
c. Age is 20 weeks

Explanation:
a. In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (6 , 12) , (8 , 15 ) , (10 , 18) , ( 12 , 21 )
First find the slope m of the line containing the two given points ((6 ,12) and (8 , 15)
m = (y₂-y₁) / (x₂-x₁)
m= (15 – 12) / (8 – 6)
m = 3/2 .
substitute the slope in the (6 ,12) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 12 = 3/2 ( x – 6)
2(y – 12) = 3(x – 6)
2y – 24 = 3x – 18
2y = 3x – 18 + 24
2y  = 3x + 6
So ,  2y  = 3x + 6 or y = \(\frac{3}{2}\)x + 3 is linear function.

b. The slope measures the rate of change of weight due to change in weeks, Here the slope of 3/2 means that as one week passes, weight of the puppy increases by 3/2 pounds.
y intercept measures the weight of the puppy, when it was born which is 3 pounds in this case measured by c.

c. put y = 33,
33 = \(\frac3}{2}\)x + 3
30 = \(\frac{3}{2}\)x
30 × 2 = 3x
x = 60/3
x = 20.
So, Age is 20 weeks.

7.4 Comparing Linear and Nonlinear Functions (pp. 295–300)
Learning Target: Understand differences between linear and nonlinear functions.

Does the table represent a linear or nonlinear function? Explain.
Question 19.

Answer: y = 3x – 8 is linear function.

Explanation:
Ordered pairs are (3 , 1 ) , (6 , 10) , (9 , 19) , (12 , 28)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (3 , 1 ) and (6 , 10)
m = (y₂-y₁) / (x₂-x₁)
m= (10 – 1) / (6– 3)
m = 9/3
m = 3
substitute the slope in the (3 , 1) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 1 = 3 ( x – 3)
y – 1 = 3x – 9
y = 3x – 9 + 1
y = 3x – 8
So ,  y = 3x – 8 is linear function.

Question 20.

Answer: y = -x + 4 is linear function.

Explanation:
Ordered pairs are (1 , 3 ) , (3 , 1) , (5 , 1) , (7 , 3)
Plot the points in the table , Draw a line through the points
First find the slope m of the line containing the two given points (1 , 3 ) and (3 , 1)
m = (y₂-y₁) / (x₂-x₁)
m= (1 – 3) / (3– 1)
m = -2/2
m = -1
substitute the slope in the (1 , 3) to get point slope to form a line.
y-y₁ = m (x-x₁)
y – 3 = -1 ( x – 1)
y – 3 = -x + 1
y = -x + 1 + 3
y = -x + 4
So ,  y = -x + 4 is linear function.

Question 21.
Does the graph represent a linear or nonlinear function? Explain.

Answer: The graph represent a non linear function.

Explanation:
As shown in the graph linear function represents a  straight line to which not happened here,
So , the graph is non linear function

Question 22.
Does the equation y = 2.3x represent a linear or nonlinear function? Explain.
Answer: y = 2.3x is a linear function.

Explanation:
Given , y = 2.3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y = 2.3(0) = 0 . co-ordinates are (0 , 0)
if x = 1 , then y = 2.3(1) = 2.3 . co-ordinates are (1 , 2.3)
if x = 2 , then y = 2.3(2) = 4.6 , co-ordinates are (2 , 4.6)
The co-ordinates (0 , 0) , (1 , 2.3) , (2 , 4.6) form a straight line .
Each x input has only one y output so it is a function .
And it  forms a straight line when graphed .
So, y = 2.3x is a linear function.

7.5 Analyzing and Sketching Graphs (pp. 301–306)
Learning Target: Use graphs of functions to describe relationships between quantities.

Question 23.
Describe the relationship between the two quantities in the graph.

Answer: At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.

Explanation:
The relationship between the graph is population and time ,
At certain point of time, line segment is increased with respect to time and maintained a constant change and again dropped gradually with the increase in time.
So, the city population is not constant at all the time.

Sketch a graph that represents the situation.
Question 24.
You climb a climbing wall. You climb halfway up the wall at a constant rate, then stop and take a break. You then climb to the top of the wall at a greater constant rate.
Answer: The graph is 

Explanation:
You start climbing a wall at a constant rate so the first portion of the graph needs to be linear with a positive slope, you then take a break which means your height is constant so the second part of the graph needs to be a horizontal line, you then start climbing again at a constant rate, so the last part of the graph needs to be linear with a positive slope.

Question 25.
The price of a stock increases at a constant rate for several months before the stock market crashes. The price then quickly decreases at a constant rate.
Answer: The graph is

Explanation:
The stock price is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, Then price begins to drop quickly so the second part of the graph needs to be linear with a steep negative slope.

Question 26.
The graph shows the sales of two companies during a particular year.

a. Describe the sales of each company.
b. Which company has greater total sales for the year?
c. Give a possible explanation for the change in each company’s sales throughout the year.
Answer: All The explanation is given below

a. Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope, and decreasing with a slight negative steep and again increasing at a constant rate increasing the sales of the company

Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales and vise versa.

b. Company A has the greater total sales for the year compared to Company B, with maintaining the sales up to the mark without losses.

c. The change in each company’s sales throughout the year,  Company A – The sales of the company is increasing at a constant rate so the first part of the graph needs to be linear with positive slope,
Company B – The sales of the company is increasing at a constant rate so the first part of the graph needs to be increase in curve with a slight decrease in the graph leading to decrease in sales

Functions Practice Test

Question 1.
List the ordered pairs shown in the mapping diagram. Then determine whether the relation is a function.

Answer: The relation is a function

Explanation:
As shown , Ordered pairs are the combinations of input and output
So , Ordered pairs are ( 2 , 9 ) , ( 4 , 9 ) , ( 6 , 10 ) , ( 8 , 11 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 2.
Draw a mapping diagram that represents the relation. Then determine whether the relation is a function. Explain.

Answer: The mapping diagram is

Explanation:
Ordered pairs from the given graph are ( -3 , 5 ) , ( -1 , 1 ) , ( -1 , 3 ) , ( 1 , 2 ) , ( 3 , 4 ) .
Each input has exactly one output ,
So , The relation is a function .

Question 3.
Write a function rule for “The output is twice the input.”
Answer: y = 2x

Explanation:
Let us say x is input and y is output , then
The output is twice the input. will be
y = 2x

Question 4.
Graph the function y = 1 – 3x.
Answer: The graph is

Explanation:
Given , y = 1 – 3x  , we know y = mx + c , where m = slope , c = constant
To obtain the graph , we should have ordered pairs ,
So , if x = 0 , then y =1 – 3(0) = 1 . co-ordinates are (0 , 1)
if x = 1 , then y = 1 – 3(1)  = -2 . co-ordinates are (1 , -2)
if x = 2 , then y = 1 – 3(2) = -5 , co-ordinates are (2 , -5)
if x = 3 , then y =1 – 3(3) = -8 , co-ordinates are (3 , -8)
The co-ordinates (0 , 1) , (1 , -2) , (2 , -5) , (3 , -8) form a straight line .

Question 5.
Use the graph to write a linear function that relates y to x.

Answer: The linear function is y = 0.5x – 1

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (-4 , -3) , (-2 , -2 ) , (0 , -1) , ( 2 , 0 )
First find the slope m of the line containing the two given points (0 , -1) and ( 2 , 0 )
m = (y₂-y₁) / (x₂-x₁)
m= (0 – (-1)) / (2 – 0)
m = 1 / 2 .
m = 0.5 .
Because the line crosses the y axis at ( 0, -1 ) , The y intercept is -1.
So , the linear function is y = 0.5x – 1 .

Question 6.
Does the table represent a linear or nonlinear function? Explain.

Answer: The linear function is y = −4x + 8

Explanation:
In order to write the function we have to write the ordered pairs of the graph ,
Ordered pairs are  (0 , 8) , (2 , 0 ) , (4 , -8) , ( 6 , -16 )
First find the slope m of the line containing the two given points (0 , 8) and (2 , 0 )
m = (y₂-y₁) / (x₂-x₁)
m= (0 – 8) / (2 – 0)
m = -4
Because the line crosses the y axis at ( 0, 8 ) , The y intercept is 8.
So , the linear function is y = −4x + 8.

Question 7.
The table shows the number of y meters a water-skier travels in x minutes.

a. Write a function that relates y to x.
b. Graph the linear function.
c. At this rate, how many kilometers will the water-skier travel in 12 minutes?
d. Another water-skier travels at the same rate but starts a minute after the first water-skier. Will this water-skier catch up to the first water-skier? Explain.
Answer: All the answers are given below

Explanation:
Ordered pairs are  (1 , 600) , (2 , 1200 ) , (3 , 1800) , ( 4 , 2400 ) , (5 , 3000)
First find the slope m of the line containing the two given points(1 , 600) and (2 , 1200 )
m = (y₂-y₁) / (x₂-x₁)
m= (1200 – 600) / (2 – 1)
m = 600
So, the line is of the form y = 600x + c
put x= 3 and y = 1800 in the above equation we get,
1800 = 600(3) + c
c = 1800 – 1800
c = 0.
So, The line is y = 600x.

b. The graph is

c. put x = 12 in y = 600x
y = 600(12)
y = 7200
7200 meters, i.e., 7.2km

d. Another water skier travels at the same rate but starts a minute after the first water skier, Since both are travelling at the same rate , the water skier who was late will always be behind the first water skier.

Question 8.
The graph shows the prices of two stocks during one day.

a. Describe the changes in the price of each stock.
b. Which stock has a greater price at the end of the day?
c. Give a possible explanation for the change in the price of Stock B throughout the day.
Answer: Detailed Explanation is given below.

Explanation:
a. The changes in the price of each stock is Stock A has the constant increase in stock for a particular time and maintains a constant price forming a straight line in the graph, and again decreasing with a negative slope and vise versa, while Stock B is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again.

b. stock B has a greater price at the end of the day, having a positive increase in the slope

c. The change in the price of Stock B throughout the day, is having steep negative slope that is decreasing in the prices with the time and having a constant horizontal line. and having a positive increase in the slope and same repeats again, compared to stock A .

Question 9.
You are competing in a footrace. You begin the race by increasing your speed at a constant rate. You then run at a constant speed until you get a cramp and have to stop. You wait until your cramp goes away before you start increasing your speed again at a constant rate. Sketch a graph that represents the situation.
Answer: The graph is

Explanation:
You begin the race by increasing your speed at a constant rate so the first portion of the graph needs to be linear with a positive slope , you then run at a constant speed so the next portion of the graph needs to be horizontal line , you then stop and take a break , so your speed is zero, which means the next portion of the line needs to be
horizontal line on the x axis , you then increase your speed again at a constant rate sop that the last portion of the graph needs to be linear with a positive slope

Functions Cumulative Practice

Question 1.
What is the slope of the line?

Answer: Not in the options but the answer is m = -4/3

Explanation:
Ordered pairs are  (-4 , 5) , (1 , -3 ),
First find the slope m of the line containing the two given points
m = (y₂-y₁) / (x₂-x₁)
m= (-3 – 5) / (2 – (-4))
m = -8/6
m = -4/3.

Question 2.
Which value of a makes the equation 24 = \(\frac{a}{3}\) – 9 true?
F. 5
G. 11
H. 45
I. 99
Answer: I. 99

Explanation:
Substitute a = 99 , in the given equation we get,
24 = \(\frac{a}{3}\) – 9
24 = \(\frac{99}{3}\) – 9
24 = 33 – 9
24 = 24.
So, last option is the correct answer.

Question 3.
A mapping diagram is shown.

What number belongs in the box so that the equation describes the function represented by the mapping diagram?

Answer: m = 7 , y = 7x + 5

Explanation:
Ordered pairs are  (4 , 33) , (7 , 54 ), (10 , 75) , (13 , 96 ),
First find the slope m of the line containing the two given points (4 , 33) and (7 , 54 )
m = (y₂-y₁) / (x₂-x₁)
m= (54 – 33) / (7 – 4)
m = 21/3
m = 7.
So, y = 7x + 5

Question 4.
What is the solution of the system of linear equations?
3x + 2y = 5
x = y + 5
A. (3, – 2)
B. (- 2, 3)
C. (- 1, 4)
D. (1, – 4)
Answer: A. (3, – 2)

Explanation:
Given 3x + 2y = 5
Then substitute , x = y + 5 in the above equation
3( y + 5) + 2y = 5
3y + 15 + 2y = 5
5y + 15 = 5
5( y + 3) = 5
y + 3 = 1
y = 1 – 3
y = -2,
substitute y = -2 in x = y + 5 then
x = 3
So, (3 , -2)

Question 5.
The director of a research lab wants to present data to donors. The data show how the lab uses a large amount of donated money for research and only a small amount of money for other expenses. Which type of display best represents these data?
F. box-and-whisker plot
G. circle graph
H. line graph
I. scatter plot
Answer: I. scatter plot

Explanation:
Scatter plot is the best graph for this type of data where vertical axis will show the amount of money and Horizontal axis will show research and other expenses.

Question 6.
Which graph shows a nonlinear function?

Answer: option B

Explanation:
As all the other options are representing the linear function that is forming a straight line  expect for option B , it is representing a non linear equation.

Question 7.
Which equation of a line passes through the point (—2, 3) and has a slope of \(\frac{3}{4}\)?

Answer:  F. y – 3 = \(\frac{3}{4}\)(x + 2)

Explanation:
Given, y – 3 = \(\frac{3}{4}\)(x + 2)
it is in the form of y = mx + c
so, slope m = \(\frac{3}{4}\)
Substitute the given points in this equation that is x = -2 and y = 3
3 – 3 = \(\frac{3}{4}\)(-2 + 2)
0 = 0.
So, F is the correct option.

Question 8.
The tables show the sales (in millions of dollars) for two companies over a five-year period.

Part A Does the first table show a linear function? Explain your reasoning.
Part B Does the second table show a linear function? Explain your reasoning.
Answer: Part A, the first table shows a linear function,
Part B the second table shows a linear function.

Explanation:
Part A – ordered pairs are (1 , 2) , (2 , 4) , (3 , 6) , (4 , 8) , (5 , 10)
Each input has exactly one output and forms a straight line when graphed
So, it is a linear function.

Part B – ordered pairs are (1 , 1) , (2 , 1) , (3 , 2) , (4 , 3) , (5 , 5)
Each input has exactly one output and does not form a straight line when graphed
So, it is a linear function.

Question 9.
The equations y = – x + 4 and y = \(\frac{1}{2}\)x – 8 form a system of linear equations. The table shows the values of y for given values of x.

What can you conclude from the table?
A. The system has one solution, when x = 0.
B. The system has one solution, when x = 4.
C. The system has one solution, when x = 8.
D. The system has no solution.
Answer: C. The system has one solution, when x = 8.

Explanation:
Given , y = – x + 4 and y = \(\frac{1}{2}\)x – 8
for x = 8 we have
y = -8 + 4 = -4
y = 0.5(8) – 8 = 4 – 8 = -4
Both the equations have one solution for x = 8
So, The system has one solution, when x = 8.

Question 10.
The vertices of a triangle are A (- 1, 3), B (1, 2), and C (- 1, – 1). Dilate the triangle using a scale factor of 2. What is the y-coordinate of the image of B?

Answer: The New right angle triangle is larger than the original one So , its a increase .

Explanation:
Given , (- 1, 3),  ( 1, 2 ),  (- 1, -1) these pairs form a right angle triangle
K = 2 , For the dilation figure multiply the 3 with the given ordered pairs , then
(- 1, 3) × 2 = ( -2 , 6)
( 1, 2) × 2 = ( 2 , 4)
(- 1, -1) × 2 = (-2 , -2)
From these new ordered pairs we form a new  right angle triangle

The New right angle triangle is larger than the original one So, it’s an increase.

Conclusion

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Big Ideas Math Book Grade K Answer Key Chapter 12 Identify Three-Dimensional Shapes

The topics covered in this chapter are Vocabulary, Two- and Three-Dimensional Shapes, Cubes and Spheres etc. So, Download Big Ideas Math Answers Grade K Chapter 12 Identify Three-Dimensional Shapes PDF for free. For more practice questions simply go to the performance task and cumulative practice which is given at the end of the chapter. Just click on the below-attached links and start your preparation from now.

Vocabulary

• Identify Three-Dimensional Shapes Vocabulary

Lesson: 1 Two- and Three-Dimensional Shapes

• Lesson 12.1 Two- and Three-Dimensional Shapes
• Two- and Three-Dimensional Shapes Homework & Practice 12.1

Lesson: 2 Describe Three-Dimensional Shapes

• Lesson 12.2 Describe Three-Dimensional Shapes
• Describe Three-Dimensional Shapes Homework & Practice 12.2

Lesson: 3 Cubes and Spheres

• Lesson 12.3 Cubes and Spheres
• Cubes and Spheres Homework & Practice 12.3

Lesson: 4 Cones and Cylinders

• Lesson 12.4 Cones and Cylinders
• Cones and Cylinders Homework & Practice 12.4

Lesson: 5 Build Three-Dimensional Shapes

• Lesson 12.5 Build Three-Dimensional Shapes
• Build Three-Dimensional Shapes Homework & Practice 12.5

Lesson: 6 Positions of Solid Shapes

• Lesson 12.6 Positions of Solid Shapes
• Positions of Solid Shapes Homework & Practice 12.6

Chapter 12: Identify Three-Dimensional Shapes

• Identify Three-Dimensional Shapes Performance Task
• Identify Three-Dimensional Shapes Activity
• Identify Three-Dimensional Shapes Chapter Practice

Identify Three-Dimensional Shapes Vocabulary

Directions:
Circle each can. Draw a square around each box. Count and write how many of each two-dimensional shape you draw.

Answer:

Answer:

Answer:

Answer:

Lesson 12.1 Two- and Three-Dimensional Shapes

Explore and Grow

Directions:
Circle any triangles, rectangles, squares, hexagons, and circles you see in the picture. Use another color to circle any objects in the picture that match the blue shapes shown. Tell what you notice about each shape.

Answer:

two-dimensional
rectangle
circle
triangle
hexagon
Total11 shapes
three-dimensional
cylinder
sphere
cube
cone
Total 8 shapes
Explanation:
A two-dimensional shape is a shape that has length and width but no depth. … A circle is one example of a two-dimensional shape. Example Two. A rectangle is another example of a two-dimensional shape.

Triangles, Rectangles, Squares, Hexagons, and Circles all these shapes are all 2-D shapes.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width, and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
All the objects in the picture represent the 3-D shapes so, they are circled with a different color.

Think and Grow

Directions:
Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

Answer:

two-dimensional
rectangle
circle
three-dimensional
cylinder
sphere
cuboid
cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Apply and Grow: Practice

Directions:
1 – 4 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

Question 1.

Answer:

two-dimensional
rectangle
circle
three-dimensional
cylinder
cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangles.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 2.

Answer:

two-dimensional
circle
square
hexagon
three-dimensional
sphere
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 3.

Answer:

two-dimensional
Triangle
three-dimensional
Cuboid
Triangle prism
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 4.

Answer:

two-dimensional
0
three-dimensional
Cylinder
Sphere
Cone

Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Think and Grow: Modeling Real Life

Directions:
Circle any shapes in the picture that are solids. Draw rectangles around any shapes in the picture that are flats. Count and write how many solids and flats you find.

three-Dimensional
________
– – – – – – – –
________

two-dimensional
________
– – – – – – – –
________

Answer:

two-dimensional
Rectangles
Total 7 flat surfaces
three-dimensional
Cubes
Sphere
Cone
Cylinder
Total 8 solids
Explanation:
Solid figures are three-dimensional. A face is a flat surface of a solid.

Two- and Three-Dimensional Shapes Homework & Practice 12.1

Directions:
1 – 3 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

Question 1.

Answer:

two-dimensional
Triangle
Square
three-dimensional
Cylinder
Sphere
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 2.

Answer:

two-dimensional
Hexagon
Circle
three-dimensional
Cube
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 3.

Answer:

two-dimensional
Rectangle
Square
three-dimensional
Cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Directions:
4 and 5 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct. 6 Circle any three-dimensional shapes in the picture. Count and write the number. Draw rectangles around any two-dimensional shapes in the picture. Count and write the number.

Question 4.

Answer:

two-dimensional
Circle
three-dimensional
Cylinder
Sphere
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 5.

Answer:

two-dimensional
Rectangle
Circle
Triangle
three-dimensional
Cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 6.

Answer:

Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Lesson 12.2 Describe Three-Dimensional Shapes

Explore and Grow

Directions:
Cut out the Roll, Stack, Slide Sort Cards. Sort the cards into the categories shown.

rolls

stacks

slides

Answer:

Think and Grow

Directions:

• Look at the solid shape on the left that rolls. Circle the other solid shapes that roll.
• Look at the solid shapes on the left that stack. Circle the other solid shapes that stack.
• Look at the solid shape on the left that slides. Circle the other solid shapes that slide.

Answer:

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.

The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere , cylinder , cone shape.

Apply and Grow: Practice

Directions:
1 Look at the solid shape on the left that rolls. Circle the other solid shapes that roll. 2 Circle the solid shapes that roll and slide. 3 Circle the solid shapes that stack and slide. 4 Circle the solid shape that does not stack or slide.

Question 1.

Answer:

Given:
The cylinder can roll,roll and slide, stack and slide
The cube can slide and stack.
The sphere can only roll.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 2.

Answer:

Given:
The cylinder can roll, stack and slide, roll and slide
Cone can roll, roll and slide.
The cube can slide and stack.
Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.

The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 3.

Answer:

Given:
A ball that represents a Sphere. The ball can only roll.
A wooden log which represent cylinder. Log can roll, roll and slide, stack and slide.
The wooden box which represents cube. The box can slide and stack.
Hat represent cone. hat can roll, roll and slide.

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 4.

Answer:

Given:
The ball which represents Sphere. The ball can only roll.
Glue stick which represents cylinder. Glue stick can roll,roll and slide, stack and slide.
The box which represents cube. Box can slide and stack.
Birthday Hat represents cone. Birthday hat can roll, roll and slide.

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Think and Grow: Modeling Real Life

Directions:
You stack the 3 objects shown. Write 1 below the object you place at the bottom of the stack, write 2 below the object you stack next, and write 3 below the object you stack last. Tell why you chose this order.

Answer:

Given:
cylinder shaped Oats box and piggy bank. Oats box and piggy bank can roll, roll and slide, stack and slide.
cube shaped Cardboard box and a Wooden box . A cardboard box and a Wooden Box can slide and stack.
cone shaped Party hats. Party hats can roll, roll and slide.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere , cylinder , cone shape.

Describe Three-Dimensional Shapes Homework & Practice 12.2

Directions:
1 Look at the solid shapes on the left that stack. Circle the other solid shapes that stack. 2 Look at the solid shape on the left that rolls. Circle the other solid shapes that roll. 3 Look at the solid shape on the left that slides. Circle the other solid shapes that slide.

Question 1.

Answer:

Explanation:
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 2.

Answer:

Explanation:
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 3.

Answer:

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.

Directions:
4 Circle the solid shapes that roll and stack. 5 Circle the solid shapes that stack and slide. 6 Circle the solid shape that does not roll. 7 You stack the 3 objects shown. Write 1 below the object you place at the bottom of the stack, write 2 below the object you stack next, and write 3 below the object you stack last. Tell why you chose this order.

Question 4.

Answer:

Cylinder shaped objects can roll and slide.

Explanation:
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 5.

Answer:

Given:
Cube shaped objects can slide and stake.
Cylinder shaped objects can roll and slide.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 6.

Answer:

Given:
Cube shaped objects can slide and stake. Cube shaped solids that does not roll.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 7.

Answer:

Explanation:
In the figure given we have 2 cylinder shaped objects and 1 cone shaped object.
Cylinders can stack, slide and roll. So, I used both the cylinder shaped objects at the bottom.
Cone can roll and slide. As cones shaped figures can not be stacked I used at the top.

Lesson 12.3 Cubes and Spheres

Explore and Grow

Directions:
Cut out the Cube and Sphere Sort Cards. Sort the cards into the categories shown.

Answer:

Think and Grow

Directions:
Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Apply and Grow: Practice

Directions:
1 Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct. 2 – 4 Circle any object that looks like a cube. Draw a rectangle around any object that looks like a sphere. Tell why your answers are correct.

Question 1.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 2.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 3.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 4.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Think and Grow: Modeling Real Life

Directions:
Use Make a Cube to build your own number cube. Draw the shape of the flat surfaces of your cube. Count and write the number of flat surfaces.

________
– – – – – – – –
________ flat surfaces
Answer:

Cubes and Spheres Homework & Practice 12.3

Directions:
1 – 3 Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct.

Question 1.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 2.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 3.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Directions:
4 – 6 Circle any object that looks like a cube. Draw a rectangle around any object that looks like a sphere. Tell why your answers are correct. 7 Draw the shape of the flat surfaces of a die. Count and write the number of flat surfaces.

Question 4.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 5.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 6.

Answer:

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 7.

Answer:

Explanation:
Dice is similar to a cube.
A cube is a region of space formed by six identical square faces joined along their edges.

Lesson 12.4 Cones and Cylinders

Explore and Grow

Directions:
Cut out the Cone and Cylinder Sort Cards. Sort the cards into the categories shown.

Answer:

Think and Grow

Directions:
Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Apply and Grow: Practice

Directions:
1 Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct. 2 – 4 Circle any object that looks like a cone. Draw a rectangle around any object that looks like a cylinder. Tell why your answers are correct.

Question 1.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 2.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 3.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 4.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Think and Grow: Modeling Real Life

Directions:
Use Make a Cylinder to build a can of vegetables. Draw the shape of the flat surfaces of your can. Count and write the number of flat surfaces.

__________
– – – – – – – – – –
__________ flat surfaces
Answer:

Cones and Cylinders Homework & Practice 12.4

Directions:
1 – 3 Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct.

Question 1.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 2.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 3.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Directions:
4 – 6 Circle any object that looks like a cone. Draw a rectangle around any object that looks like a cylinder. Tell why your answers are correct. 7 Draw the shape of the flat surface of a cone. Count and write the number of flat surfaces.

Question 4.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 5.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 6.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 7.

Answer:

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

Lesson 12.5 Build Three-Dimensional Shapes

Explore and Grow

Directions:
Use your materials to build one of the three-dimensional shapes shown. Circle the three-dimensional shape that you build.

Answer:

Think and Grow