{"id":90189,"date":"2022-03-31T15:18:52","date_gmt":"2022-03-31T09:48:52","guid":{"rendered":"https:\/\/bigideasmathanswers.com\/?p=90189"},"modified":"2022-04-07T14:31:32","modified_gmt":"2022-04-07T09:01:32","slug":"envision-math-common-core-grade-8-answer-key-topic-3","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/envision-math-common-core-grade-8-answer-key-topic-3\/","title":{"rendered":"enVision Math Common Core Grade 8 Answer Key Topic 3 Use Functions To Model Relationships"},"content":{"rendered":"

Practice with the help of enVision Math Common Core Grade 8 Answer Key<\/a> Topic 3 Use Functions to Model Relationships<\/strong> regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 8th Grade Answers Key Topic 3 Use Functions To Model Relationships<\/h2>\n

Topic 3 GET READY!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.<\/p>\n

Question 1.
\nThe ____ is the ratio of the vertical change to the horizontal change of a line.
\nAnswer:
\nWe know that,
\nThe “Slope” is the ratio of the vertical change to the horizontal change of a line
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Slope”<\/p>\n

Question 2.
\nA relationship that can be modeled by the equation y = mx is a ___
\nAnswer:
\nWe know that,
\nA relationship that can be modeled by the equation y = mx is a “Proportional relationship”
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Proportional relationship”<\/p>\n

Question 3.
\ny-value at which a line of a graph crosses the y-axis is called the ___
\nAnswer:
\nWe know that,
\ny-value at which a line of a graph crosses the y-axis is called the “y-intercept”
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “y-intercept”<\/p>\n

Question 4.
\nAn equation written in the form y = mx + b is called the ___
\nAnswer:
\nWe know that,
\nAn equation written in the form y = mx + b is called the “Linear equation” or the “Slope-intercept form”
\nHence, from the above,
\nWe can conclude that the best term from the box to complete the given definition is “Linear equation”(or) the “Slope-intercept form”<\/p>\n

Slope and y-Intercept<\/strong><\/p>\n

Find the slope and y-intercept of a line that passes through these points.<\/p>\n

Question 5.
\n(2, 2) and (3, 0)
\nAnswer:
\nThe given points are:
\n(2, 2), and (3, 0)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>,y2<\/sub>)
\nWe know that,
\nSlope(m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nm = \\(\\frac{0 – 2}{3 – 2}\\)
\n= \\(\\frac{-2}{1}\\)
\n= -2
\nWe know that,
\nThe linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nWe know that,
\nWe can obtain the y-intercept by putting the value of x equal to 0
\nSo,
\ny = -2x + c
\nSubstitute (3, 0) or (2, 2) in the above equation
\nSo,
\n0 = -6 + c
\nSo,
\nc = 6
\nHence, from the above,
\nWe can conclude that
\nThe slope of a line that passes through the given points is: -2
\nThe y-intercept of a line is: 6<\/p>\n

Question 6.
\n(1, 5) and (4, 10)
\nAnswer:
\nThe given points are:
\n(1, 5), and (4, 10)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>,y2<\/sub>)
\nWe know that,
\nSlope(m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nm = \\(\\frac{10 – 5}{4 – 1}\\)
\n= \\(\\frac{5}{3}\\)
\nWe know that,
\nThe linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nWe know that,
\nWe can obtain the y-intercept by putting the value of x equal to 0
\nSo,
\ny = \\(\\frac{5}{3}\\)x + c
\nSubstitute (4, 10) or (1, 5) in the above equation
\nSo,
\n5 = \\(\\frac{5}{3}\\) + c
\nSo,
\nc = \\(\\frac{10}{3}\\)
\nHence, from the above,
\nWe can conclude that
\nThe slope of a line that passes through the given points is: \\(\\frac{5}{3}\\)
\nThe y-intercept of a line is: \\(\\frac{10}{3}\\)<\/p>\n

Question 7.
\n(8, 2) and (-8,6)
\nAnswer:
\nThe given points are:
\n(8, 2), and (-8, 6)
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>,y2<\/sub>)
\nWe know that,
\nSlope(m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nm = \\(\\frac{6 – 2}{-8 – 8}\\)
\n= \\(\\frac{4}{-16}\\)
\n= –\\(\\frac{1}{4}\\)
\nWe know that,
\nThe linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nWe know that,
\nWe can obtain the y-intercept by putting the value of x equal to 0
\nSo,
\ny = –\\(\\frac{1}{4}\\)x + c
\nSubstitute (-8, 6) or (8, 2) in the above equation
\nSo,
\n2 = –\\(\\frac{1}{4}\\) (8) + c
\nSo,
\nc = 4
\nHence, from the above,
\nWe can conclude that
\nThe slope of a line that passes through the given points is: –\\(\\frac{1}{4}\\)
\nThe y-intercept of a line is: 4<\/p>\n

Compare Proportional Relationships<\/strong><\/p>\n

Jenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6. Which store sells the drinks for less? Use the graph to compare the unit costs of the drinks.<\/p>\n

Question 8.
\n\"Envision
\nAnswer:
\nJenna’s mother is shopping for energy drinks in 12-ounce bottles for Jenna’s soccer team. Store A sells a case of 18 bottles for $10. Store B sells a case of 12 bottles for $6
\nNow,
\nThe unit cost rate of a bottle in store A = \\(\\frac{The cost of 18 bottles}{18}\\)
\n= \\(\\frac{$10}{18}\\)
\n= $0.55
\nThe unit cost rate of a bottle in store B = \\(\\frac{The cost of 12 bottles}{12}\\)
\n= \\(\\frac{$6}{12}\\)
\n= $0.5
\nSo,
\nThe representation of the unit cost rate of a bottle in both stores is:
\n\"\"
\nSo,
\nFrom the above graph,
\nWe can observe that
\nThe unit cost rate of a bottle in store B < The unit cost rate of a bottle in store A
\nHence, from the above,
\nWe can conclude that store B sells the drinks for less cost<\/p>\n

Linear Equations<\/strong><\/p>\n

Question 9.
\nWrite the equation for the graph of the line shown.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe y-intercept is the value of the point that passes through the y-axis
\nSo,
\nFrom the graph,
\nThe point that passes through the y-axis is: (0, -6)
\nSo,
\nThe y-intercept is: -6
\nNow,
\nTo find the slope,
\nThe points are: (2, 2), and (0, -6)
\nNow,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>,y2<\/sub>)
\nWe know that,
\nSlope(m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nSo,
\nm = \\(\\frac{-6 – 2}{0 – 2}\\)
\n= \\(\\frac{-8}{-2}\\)
\n= 4
\nWe know that,
\nThe linear equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = 4x – 6
\nHence, from the above,
\nWe can conclude that the equation of the line for the given graph is:
\ny = 4x – 6<\/p>\n

Language Development<\/strong><\/p>\n

Write key words or phrases associated with each representation. Then write function or not a function on the given lines.
\n\"Envision
\nAnswer:
\n\"\"<\/p>\n

Topic 3 PICK A PROJECT<\/h3>\n

PROJECT 3A<\/strong>
\n\"Envision
\nWhat machine could be invented to make your life better?
\nPROJECT: BUILD A RUBE GOLDBERG MACHINE<\/p>\n

PROJECT 3B<\/strong>
\n\"Envision
\nWhat games can you play indoors?
\nPROJECT: MAKE A MATH
\nCARD GAME<\/p>\n

PROJECT 3C
\n<\/strong>\"Envision
\nWhat are the steps for fixing a leaky pipe?
\nPROJECT: PLAN A MAINTENANCE ROUTE<\/p>\n

PROJECT 3D
\n<\/strong>\"Envision
\nIf you were to make a video game, what kind of game would it be?
\nPROJECT: DESIGN A VIDEO GAME ELEMENT<\/p>\n

Lesson 3.1 Understand Relations and Functions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

The 10 members of Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
\nWhose plan would you recommend? Explain.
\n\"Envision
\nRAFFLE TICKETS
\n$1 1 ticket
\n$5 6 tickets
\n$20 25 tickets
\nAnswer:
\nIt is given that
\nThe 10 members of the Photography Club want to raise $500, so they will hold a raffle with donated prizes. Jesse proposes that to reach their goal, each member should sell 50 raffle tickets. Alexis proposes that each member should raise $50.
\nIt is also given that
\n$1\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a01 ticket
\n$5\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 6 tickets
\n$20\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 –\u00a0 \u00a0 \u00a0 \u00a0 25 tickets
\nNow,
\nAccording to Jesse’s goal,
\nEach member should sell 50 raffle tickets to make the total amount of $500
\nThe possible combinations may be:
\n50 $1 tickets will be sold by each member
\nAny other combination is not possible
\nAccording to Alexis’s plan,
\nEach member should raise $50 to make the total amount of $500
\nIt is possible and very easy because $50 by each member can be raised in many ways
\nHence, from the above,
\nWe can conclude that Alexis’s plan would be recommended<\/p>\n

Focus on math practices<\/strong>
\nReasoning How are the two plans different? How are they similar?
\nAnswer:
\nAccording to Jesse’s goal,
\nEach member should sell 50 raffle tickets to make the total amount of $500
\nThe possible combinations may be:
\n50 $1 tickets will be sold by each member
\nAny other combination is not possible
\nNow,
\nAccording to Alexis’s plan,
\nEach member should raise $50 to make the total amount of $500
\nIt is possible and very easy because $50 by each member can be raised in many ways<\/p>\n

? Essential Question<\/strong>
\nwhen is a relation a function?
\nAnswer:
\nA “Relation” from a set X to a set Y is called a “Function” if and only if each element of X is related to exactly one element in Y.<\/p>\n

Try It!<\/strong>
\nJoe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
\n(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
\n\"Envision
\nComplete the arrow diagram. Is the area of a brochure a function of the side length? Explain.
\nAnswer:
\nJoe needs to advertise his company. He considers several different brochures of different side lengths and areas. He presents the data as ordered pairs (side length, area).
\n(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
\nWe know that,
\nThe ordered pairs can be represented in the form of (x, y)
\nWhere,
\nx is the input
\ny is the output
\nNow,
\nFrom the given ordered pairs,
\nWe can observe that for different values of the input, there are different values of output
\nNote:
\nIf there are the same outputs for the different inputs, then also a relationship is considered a function
\nSo,
\nThe complete arrow diagram for the given ordered pairs are:
\n\"\"
\nHence, from the above,
\nWe can conclude that the area of the brochure is a function of the side length<\/p>\n

Convince Me!<\/strong>
\nThere are two outputs of 24. Does this help you determine whether the relation is a function? Explain.
\nAnswer:
\nWe know that,
\nIf there are the same outputs for the different inputs, then also a relationship is considered a function
\nHence, from the above
\nWe can conclude that even for the two outputs of 24, the given relationship is considered a function<\/p>\n

Try It!<\/strong><\/p>\n

Frank reverses the ordered pairs to show the heights and ages of the same six students. Is age a function of height? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nFrank reverses the ordered pairs to show the heights and ages of the same six students.
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nFrom the given table,
\nWe can observe that
\nFor different values of age, there are different values of height
\nWhere,
\nAge —-> Input
\nHeight —> Output
\nHence, from the above,
\nWe can conclude that age is a function of height<\/p>\n

Try It!<\/strong>
\nHeather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking. Is Heather correct? Explain.
\nAnswer:
\nIt is given that
\nHeather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking.
\nNow,
\nThe table for the given situation is: (Example 3)
\n\"\"
\nNow,
\nFrom the table,
\nWe can observe that
\nThere are different costs for the different times in hours
\nSo,
\nWe can say that cost is a function of time
\nHence, from the above,
\nWe can conclude that Heather’s claim is correct<\/p>\n

KEY CONCEPT<\/strong>
\nA relation is a function if each input corresponds to exactly one output. You can use an arrow diagram or a table to determine whether a relation is a function.
\nThis relation is a function.
\n\"Envision
\nThis relation is not a function.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nwhen is a relation a function?
\nAnswer:
\nA\u00a0relation from a set X to a set Y is called a function if and only if each element of X is related to exactly one element in Y<\/p>\n

Question 2.
\nModel with Math<\/strong>
\nHow can you use different representations of a relation to determine whether the relation is a function?
\nAnswer:
\nRelations can be displayed as a table, a mapping, or a graph. In a table, the x-values and y-values are listed in separate columns. Each row represents\u00a0an ordered pair: Displaying a\u00a0relation\u00a0as a table<\/p>\n

Question 3.
\nGeneralize<\/strong>
\nIs a relation always a function? Is a function always a relation? Explain.
\nAnswer:
\nAll functions are relations, but not all relations are functions. A function is a relation that for each input, there is only one output. Here are mappings of functions. The domain is the input or the x-value, and the range is the output or the y-value.<\/p>\n

Question 4.
\nIs the relation shown below a function? Explain.
\n\"Envision
\nAnswer:
\nThe given relation is:
\n\"Envision
\nFrom the given relation,
\nWe can observe that there is the same input for the different outputs,
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is not a function<\/p>\n

Question 5.
\nIs the relation shown below a function? Explain.
\n\"Envision
\nAnswer:
\nThe given relation is:
\n\"Envision
\nFrom the given relation,
\nWe can observe that there are different outputs for different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 6.
\nIs the relation shown below a function? Explain.
\n(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
\nAnswer:
\nThe given relation is:
\n(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
\nFrom the given relation,
\nWe can observe that there are different outputs for different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nThe set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
\na. Make an arrow diagram that represents
\nAnswer:
\nThe arrow diagram for the given relation is:
\n\"\"
\nb. is the relation a function? Explain.
\nAnswer:
\nIt is given that
\nThe set of ordered pairs (1, 19), (2, 23), (3, 23), (4, 29), (5, 31) represents the number of tickets sold for a fundraiser. The input values represent the day and the output values represent the number of tickets sold on that day.
\nNow,
\nThe given relation is:
\n(1, 19), (2, 23), (3, 23), (4, 29), (5, 31)
\nFrom the given relation,
\nWe can observe that there are different outputs for different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 8.
\nDoes the relation shown below represent a function? Explain.
\n(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
\nAnswer:
\nThe given relation is:
\n(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
\nFrom the given relation,
\nWe can observe that there are different outputs for different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 9.
\nIs the relation shown in the table a function? Explain.
\n\"Envision
\nAnswer:
\nThe given relation is:
\n\"Envision<\/p>\n

From the given relation,
\nWe can observe that there are different outputs for the same inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is not a function<\/p>\n

Question 10.
\nConstruct Arguments<\/strong>
\nDuring a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in \u00b0C).
\n(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100) Is the relation a function? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nDuring a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in \u00b0C).
\n(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
\nWe know that,
\nAn ordered pair can be represented as (x, y)
\nWhere,
\nx is the time
\ny is the temperature in \u00b0C
\nNow,
\nThe given relation is:
\n(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
\nFrom the given relation,
\nWe can observe that there are different outputs for the different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 11.
\nReasoning<\/strong>
\nTaylor has tracked the number of students in his grade since third grade. He records his data in the table below. Is the relation a function? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nTaylor has tracked the number of students in his grade since third grade. He records his data in the table
\nNow,
\nThe given table is:
\n\"Envision
\nFrom the given table,
\nWe can observe that there are different outputs (People) for the different inputs (Grade)
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given table is a function<\/p>\n

Question 12.
\nJames raises chickens. He tracks the number of eggs his chickens lay at the end of each week. Is this relation a function? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nJames raises chickens. He tracks the number of eggs his chickens lay at the end of each week
\nNow,
\nThe given relation is:
\n\"Envision
\nFrom the given relation,
\nWe can observe that there are different outputs (Eggs) for the different inputs (Weeks)
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 13.
\nRelations P and Q are shown below.
\n\"Envision
\na. Make an arrow diagram to represent Relation P.
\nAnswer:
\nThe given relation is:
\n\"Envision
\nHence,
\nThe arrow diagram to represent the relation P is:
\n\"\"
\n\"Envision
\nb. Make an arrow diagram to represent Relation Q.
\nAnswer:
\nThe given relation is:
\n\"Envision
\nHence,
\nThe arrow diagram to represent the relation Q is:
\n\"\"<\/p>\n

c. Which relation is a function? Explain.
\nAnswer:
\nFrom relation P,
\nWe can observe that there are different outputs for the different inputs
\nFrom relation Q,
\nWe can observe that there are different outputs for the same inputs
\nHence, from the above,
\nWe can conclude that relation P is a function<\/p>\n

Question 14.
\nHigher-Order Thinking<\/strong>
\nOn a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function. What error did Bobby most likely make?
\n\"Envision
\nAnswer:
\nIt is given that
\nOn a recent test, students had to determine whether the relation represented by the ordered pairs (1, 2), (6, 12), (12, 24), (18, 36) is a function. Bobby drew the arrow diagram on the right and said the relationship was not a function.
\nNow,
\nFrom the given arrow diagram and ordered pairs,
\nWe can observe that
\nIn the arrow diagram, inputs and outputs are reversely represented
\nHence, from the above,
\nWe can conclude that the error Bobby most likely made is the reversal of inputs and outputs<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 15.
\nWrite the set of ordered pairs that is represented by the arrow diagram at the right. Is the relation a function? Explain.
\n\"Envision
\nAnswer:
\nThe given arrow diagram is:
\n\"Envision
\nIn the arrow diagram,
\nThe left side represented the inputs and the right side represented the outputs
\nSo,
\nThe representation of the arrow diagram in the form of the ordered pairs (Input, Output) are:
\n(49, 13), (61, 36), (10, 27), (76, 52), (23, 52)
\nFrom the above relation,
\nWe can observe that there are different outputs for the different inputs
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 16.
\nWhich of these relations are functions? Select all that apply.
\n\"Envision
\nAnswer:
\nWe know that,
\nA relation can be considered as a function when the different inputs have different outputs
\nSo,
\nFrom the given relations,
\nRelation 2, Relation 3 are the functions
\nHence, from the above,
\nWe can conclude that Relation 2 and Relation 3 are the functions<\/p>\n

Lesson 3.2 Connect Representations of Functions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Eliza volunteers at a nearby aquarium, where she tracks the migratory patterns of humpback whales from their feeding grounds to their breeding grounds. She recorded the distance, in miles, traveled by the whales each day for the first 7-day period of their migration. Based on Eliza’s data, how long will it take the humpback whales to travel the 3,100 miles to their breeding grounds?
\n\"Envision<\/p>\n

Focus on math practices<\/strong>
\nConstruct Arguments<\/strong> How does finding an average distance the whales travel in miles help with finding a solution to this problem?<\/p>\n

? Essential Question<\/strong>
\nWhat are different representations of a function?
\nAnswer:
\nRelationships and\u00a0functions\u00a0can be represented as graphs, tables, equations, or verbal descriptions. Each\u00a0representation\u00a0gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.<\/p>\n

Try It!<\/strong>
\nAs the pump is pumping water, the amount of water in the pool decreases at a constant rate. Complete the statements below. Then graph the function.
\n\"Envision
\nThe amount of water remaining in the pool is \"Envision gallons.
\nThe amount of water pumped each hour is \"Envision gallons.
\nThe equation is \"Envision
\nAnswer:
\nIt is given that
\nAs the pump is pumping water, the amount of water in the pool decreases at a constant rate
\nNow,
\nLet the initial amount of water present in the pool is: 9,000 gallons
\nSo,
\nThe rate of the amount of water that pumped each hour = \\(\\frac{The initial amount of water present in the pool}{The time that is present where the initial amount of water present}\\)
\n= \\(\\frac{9,000}{12}\\)
\n= 750 gallons per hour
\nSo,
\nThe amount of water remaining in the pool = The initial amount of water present in the pool – The amount of water that pumped each hour
\n= 9,000 – 1,500
\n= 7,500 gallons
\nLet the number of hours be x
\nWe know that,
\nThe linear equation is in the form of
\ny = mx + c
\nSo,
\nThe total amount of water present in the pool = The rate at which the water pumps out + The amount of water that pumped each hour
\n9,000 = 750x + 7,500
\nHence, from the above,
\nWe can conclude that
\nThe amount of water remaining in the pool is 7,500 gallons.
\nThe amount of water pumped each hour is 1,500 gallons.
\nThe equation is:
\n9,000 = 750x + 7,500<\/p>\n

Convince Me!<\/strong>
\nHow is the rate of change of this function different from that in Example 1? Explain.
\nAnswer:
\nThe rate of change of the function present in Example 1 is increasing at a constant rate whereas the rate of change of the function in this situation is decreasing at a constant rate<\/p>\n

Try It!<\/strong>
\nDraw a graph that represents a linear function?
\n\"Envision
\nAnswer:
\nWe know that,
\nThe representation of the linear equation is:
\ny = mx —–> Slope form
\ny = mx + c —-> Slope-intercept form
\nNow,
\nLet the linear equation in the slope-intercept form be:
\ny = x + 3
\nHence,
\nThe graph of the above linear equation in the coordinate plane is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can represent a function in different ways: in a table, in a graph, or as an equation.
\nA day at the amusement park costs $10 for an entrance fee and $2.50 for each ride ticket.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> What are different representations of a function?
\nAnswer:
\nRelationships and\u00a0functions\u00a0can be represented as graphs, tables, equations, or verbal descriptions. Each\u00a0representation\u00a0gives us certain information. A table of values, mapping diagram, or set of ordered pairs gives us a list of input values and their corresponding output values.<\/p>\n

Question 2.
\nUse Appropriate Tools<\/strong> How can you use a graph to determine that a relationship is NOT a function?
\nAnswer:
\nUse\u00a0the vertical line test to\u00a0determine\u00a0whether or\u00a0not\u00a0a\u00a0graph\u00a0represents a\u00a0function. If a vertical line is moved across the\u00a0graph\u00a0and, at any time, touches the\u00a0graph\u00a0at only one point, then the\u00a0graph\u00a0is a\u00a0function. If the vertical line touches the\u00a0graph\u00a0at more than one point, then the\u00a0graph\u00a0is\u00a0not a function.<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
\nAnswer:
\nThe points can be connected by a straight line. Thus, the ordered pairs represent a linear function.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nEach week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock. Is the relationship a linear function? Use the graph below to support your answer.
\n\"Envision
\nAnswer:
\nIt is given that
\nEach week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock.
\nWe know that,
\nA relation is defined as a function only when there are different outputs for different inputs
\nNow,
\nFrom the given table,
\nWe can observe that the outputs (party hats) are different for different inputs (Weeks)
\nNow,
\nThe given function can be called “Linear function” if the rate of change is constant
\nThe given function can be called a “Non-linear function” if the rate of change is not constant
\nNow,
\nThe rate of change = Initial value – Next value
\nHence, from the above,
\nWe can conclude that the given relationship is a linear function<\/p>\n

Question 5.
\nHow can Darlene use the graph above to know when to order more party hats?
\nAnswer:
\nFrom the graph,
\nDarlene know when to order more hats when there is no stack when observing the track sheet<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Leveled Practice<\/strong> In 6-7, explain whether each graph represents a function.<\/p>\n

Question 6.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the above graph,
\nWe can observe that each input has a different output and the rate of change is constant
\nHence, from the above,
\nWe can conclude that the given graph represents a function<\/p>\n

Question 7.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that each input has a different output but the rate of change is not constant
\nHence, from the above,
\nWe can conclude that the given graph represents a function but a non-linear function<\/p>\n

Question 8.
\nHannah approximates the areas of circles using the equation A = 3r2<\/sup>\u00a0and records areas of circles with different radius lengths in a table.
\n\"Envision
\na. Graph the ordered pairs from the table.
\nAnswer:
\nThe given table is:
\n\"Envision
\nSo,
\nFrom the table,
\nThe representation of the ordered pairs (in, in\u00b2) are:
\n(1, 3), (2, 12), (3, 27), (4, 48), (5, 75)
\nHence,
\nThe representation of the ordered pairs in the coordinate plane is:
\n\"\"<\/p>\n

b. Is the relation a function? Explain.
\nAnswer:
\nFrom part (a),
\nWe can observe that there are different outputs for different inputs
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 9.
\nModel with Math<\/strong>
\nThe relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph. is the perimeter of the figure a function of the number of hexagons? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph
\nNow,
\nFrom the graph,
\nWe can observe that for different values of perimeters, the number of hexagons is also different
\nHence, from the above,
\nWe can conclude that the perimeter of a figure is a function of the number of hexagons<\/p>\n

Question 10.
\nConstruct Arguments<\/strong> Do the ordered pairs plotted in the graph below represent a function? Explain.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that for the different values of x, there are different values of y
\nHence, from the above,
\nWe can conclude that the given ordered pairs in the graph represents a function<\/p>\n

Question 11.
\nA train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
\na. Write a function that relates the distance traveled, d, to the time, t.
\n\"Envision
\nAnswer:
\nIt is given that
\nA train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nHere,
\nSpeed is constant
\nSo,
\nDistance = m (Time)
\nd = mt
\nWhere,
\nm is the rate of change or proportionality constant
\nNow,
\nm = \\(\\frac{Distance}{Time}\\)
\n= \\(\\frac{360}{3}\\)
\n= 120 km \/ hour
\nHence, from the above,
\nWe can conclude that the function that relates to the distance d, and time t is:
\nd = 120t<\/p>\n

b. Graph the function and tell whether it is a linear function or a nonlinear function.
\nThe function is a \"Envision function.
\nAnswer:
\nFrom part (a),
\nThe function that relates the distance d, and time t is:
\nd = 120t
\nNow,
\nCompare the above equation with y = mx
\nHence,
\nThe graph of the above function is:
\n\"\"
\nHence, from the above,
\nWe can conclude that the given function is a linear function from the above graph<\/p>\n

Question 12.
\nHigher-Order Thinking<\/strong> Tell whether each graph is a function and justify your answer. Which graph is not a good representation of a real-world situation? Explain.
\n\"Envision
\nAnswer:
\nThe given graphs are:
\n\"Envision
\nFrom graph A,
\nWe can observe that there are different outputs for the same input
\nSo,
\nGraph A does not represent the function
\nFrom graph B,
\nWe can observe that there are different outputs for the different inputs
\nSo,
\nGraph B does represent a function
\nHence, from the above,
\nWe can say that graph A does not represent the real-world situation<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 13.
\nYou have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
\nPART A<\/strong>
\nComplete the table.
\n\"Envision
\nAnswer:
\nIt is given that
\nYou have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
\nHence,
\nThe completed table for the given situation is:
\n\"\"
\nPART B<\/strong>
\nIs the relation a function? If so, is it a linear function or a nonlinear function? Explain.
\n\"Envision
\nAnswer:
\nFrom part (a),
\nWe can observe that the ant population is different for the different number of months
\nNow,
\nRate of change = \\(\\frac{44}{22}\\)
\n= 2
\nSo,
\nThe rate of change is also constant for all the table
\nHence, from the above,
\nWe can conclude that the given relation is a linear function<\/p>\n

Question 14.
\nUse the function y = \\(\\frac{3}{2}\\)x + 3 to complete the table of values.
\n\"Envision
\nAnswer:
\nThe given function is:
\ny = \\(\\frac{3}{2}\\)x + 3
\nHence,
\nThe completed table for the given values of y is:
\n\"\"<\/p>\n

Lesson 3.3 Compare Linear and Nonlinear Functions<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads below. What do the two plans have in common? How are they different? When is Movies4You a better deal than Family Stream?
\n\"Envision
\nAnswer:
\nIt is given that
\nTwo streaming video subscription services offer family plans with different monthly costs, as shown in the ads
\nNow,
\nLet the number of devices be x
\nLet the total subscription cost be y
\nSo,
\nFor Movies 4 U,
\nThe total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
\ny = $10 + $2x
\nFor Family Stream,
\nThe total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
\ny = \\(\\frac{$12}{4}\\) + $1x
\ny = $3 + $1x
\nNow,
\nThe above equations are in the form of slope-intercept form
\nWe know that,
\nThe slope-intercept form of the equation is:
\ny = mx + c
\nNow,
\nThe common points in the two plans are:
\nA) The subscription cost of the first device plan
\nB) The additional fee plan
\nThe different points in the two plans are:
\nA) The first plan consists of the additional fee of each device
\nB) The second plan consists of the additional fee for greater than 5 devices
\nNow,
\nMovies 4 U is better than Family Stream when the subscription cost of the first device will be less<\/p>\n

Model with Math<\/strong>
\nHow can you represent the relationship between cost and number of devices?
\n<\/span>Answer:
\nThe representation of the relationship between the cost and the number of devices is:
\nThe total subscription cost = The subscription cost of the first device + The subscription fee for additional devices<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships<\/strong> Describe the relationship between the cost and the number of devices for each service. What do you notice about each relationship?
\nAnswer:
\nThe relationship between the cost and the number of devices for each service is:
\nThe total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
\nIn the service of Family Stream,
\nThe subscription fee is given for up to 4 devices and the additional subscription fee is for greater than 5 devices<\/p>\n

? Essential Question<\/strong>
\nHow can you compare two functions?
\nAnswer:
\nThe two functions can be compared by:
\nA) Identify the rate of change for the first function
\nB) Identify the rate of change for the second function
\nC) Identify the y-intercept of the first function
\nD) Identify the y-intercept of the second function
\nE) Compare the properties of each function<\/p>\n

Try It!<\/strong>
\nThe welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks. How does it compare to the other two?
\nAnswer:
\nIt is given that
\nThe welding rate of a third robot is represented by the equation t = 10.8w, where t represents the time in minutes and w represents the number of welding tasks (Refer to Example 1)
\nNow,
\nFor a third robot,
\nThe wielding rate = \\(\\frac{The number of wielding tasks (w)}{Time in minutes (t)}\\)
\n= 10.8 (From the equation t = 10.8w)
\nNow,
\nWhen we compare the wielding rates of the three robots,
\nThe wielding rate of the first robot (10.4) < The wielding rate of the third robot (10.8) < The wielding rate of the second robot (11.2)
\nHence, from the above,
\nWe can conclude that the comparison of the wielding rates of the three robots is:
\nThe first robot’s wielding rate < The third robot’s wielding rate < The second robot’s wielding rate<\/p>\n

Convince Me!<\/strong>
\nHow can linear equations help you compare linear functions?
\nAnswer:
\nWhile all\u00a0linear equations\u00a0produce straight lines when graphed, not all\u00a0linear equations\u00a0produce\u00a0linear functions. In order\u00a0to\u00a0be a\u00a0linear function, a graph must be both\u00a0linear\u00a0(a straight line) and a\u00a0function\u00a0(matching each x-value\u00a0to\u00a0only one y-value).<\/p>\n

Try It!<\/strong>
\nCompare the properties of these two linear functions.
\n\"Envision
\nAnswer:
\nThe given functions are:
\n\"Envision
\nNow,
\nFor function 1,
\nFind out the rate of change and the y-intercept i.e., the initial value
\nWe know that,
\nRate of change = \\(\\frac{y}{x}\\)
\nNow,
\nFor y = 1 and x = 2.
\nRate of change = 0.5
\nFor y = 5.5 and x = 5,
\nRate of change = 1.1
\nSO,
\nFrom the above values,
\nWe can say that the rate of change is not constant
\nSo,
\nThe given function is a non-linear function and it does not have any initial value i.e., the y-intercept is 0
\nNow,
\nFor function 2,
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the slope or the rate of change
\nc is the initial value or the y-intercept
\nSo,
\nFrom the given equation,
\nRate of change (m): 2
\nThe y-intercept is: -4
\nHence, from the above 2 functions,
\nWe can conclude that
\nThe y-intercept of function1 > The y-intercept of function 2<\/p>\n

KEY CONCEPT<\/strong>
\nYou can compare functions in different representations by using the properties of functions.
\nCompare the constant rate of change and the initial value.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong>
\nHow can you compare two functions?
\nAnswer:
\nThe two functions can be compared by:
\nA) Identify the rate of change for the first function
\nB) Identify the rate of change for the second function
\nC) Identify the y-intercept of the first function
\nD) Identify the y-intercept of the second function
\nE) Compare the properties of each function<\/p>\n

Question 2.
\nReasoning<\/strong> Anne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table. How can you determine who is running faster?
\n\"Envision
\nAnswer:
\nIt is given that
\nAnne is running on a trail at an average speed of 6 miles per hour beginning at mile marker 4. John is running on the same trail at a constant speed, shown in the table.
\nSo,
\nFor Anne,
\nThe rate of change is defined as the average speed
\nThe y-intercept is defined as the beginning point
\nHence,
\nFor Anne,
\nThe rate of change is: 6 miles per hour
\nThe y-intercept is: 4
\nNow,
\nThe given table is:
\n\"Envision
\nFrom the given table,
\nFor John,
\nRate of change = \\(\\frac{y}{x}\\) = \\(\\frac{Mile marker}{Time (hours)}\\)
\nNow,
\nThe total distance traveled by John (y) = Final value – Initial value
\n= 11.5 – 1
\n= 10.5 miles
\nThe total time took by John (x) = 1.5 hours
\nSo,
\nRate of change = \\(\\frac{y}{x}\\)
\n= \\(\\frac{10.5}{1.5}\\)
\n= 7 miles per hour
\nNow,
\nWe know that,
\nThe y-intercept is the value of y when x = 0
\nSo,
\nThe y-intercept is: 1
\nHence, from the above,
\nBy comparing the values of the rate of change,
\nWe can conclude that John is running faster<\/p>\n

Question 3.
\nReasoning<\/strong> In Item 2, how do Anne and John’s starting positions compare? Explain.
\nAnswer:
\nWe know that,
\nThe starting positions are nothing but the initial positions of both Anne and John i.e., the y-intercepts of both Anne and John
\nSo,
\nThe y-intercept of Anne is: 4
\nThe y-intercept of John is: 1
\nHence, from the above,
\nBy comparing the y-intercepts,
\nWe can conclude that
\nThe starting position of Anne > The starting position of John<\/p>\n

Do You Know How?<\/strong><\/p>\n

Felipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.
\n\"Envision
\nAnswer:
\nIt is given that
\nFelipe and Samantha use a payment plan to buy musical instruments. Felipe writes the equation y = -30x + 290 to represent the amount owed, y, after x payments. The graph shows how much Samantha owes after each payment.
\nNow,
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the rate of change
\nc is the initial value or the y-intercept
\nSo,
\nFrom the given equation
\nFor Felipe,
\nThe rate of change is: -30
\nThe initial value is: 290
\nNow,
\nThe given graph is:
\n\"Envision
\nFor Samantha,
\nFrom the given graph,
\nThe initial value is: 240
\nThe rate of change = \\(\\frac{Any value of y}{The value of x that corresponds to the value of y}\\)
\n= \\(\\frac{120}{6}\\)
\n= 20<\/p>\n

Question 4.
\nWhose musical instrument costs more, Felipe’s or Samantha’s? Explain.
\nAnswer:
\nWe know that,
\nIf the amount owed (y) is less i.e, the rate of change is negative, then the cost of the instrument will also be less
\nSo,
\nWhen we compare the rate of change of Felip and Samantha,
\nThe rate of change of Felip < The rate of change of Samantha
\nHence, from the above comparison,
\nWe can conclude that the instrument of Samantha costs more<\/p>\n

Question 5.
\nWho will pay more each month? Explain.
\nAnswer:
\nWhen we compare the rate of change of Felip and Samantha,
\nThe rate of change of Felip < The rate of change of Samantha
\nHence, from the above comparison,
\nWe can conclude that Samantha will pay more each month<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nTwo linear functions are shown below. Which function has the greater rate of change?
\n\"Envision
\nAnswer:
\nThe given linear functions are:
\n\"Envision
\nWe know that,
\nThe rate of change = \\(\\frac{y}{x}\\)
\nSo,
\nFor Function A,
\nThe rate of change = \\(\\frac{Any value of y}{The value of x that corresponds to y}\\)
\n= \\(\\frac{4}{2}\\)
\n= 2
\nFor Function B,
\nThe rate of change = \\(\\frac{y}{x}\\)
\n=\u00a0 \\(\\frac{3}{2}\\)
\n= 1.5
\nHence, from the above,
\nWe can conclude that Function A has a greater rate of change<\/p>\n

Question 7.
\nTwo linear functions are shown below. Which function has the greater initial value?
\n\"Envision
\nAnswer:
\nThe given functions are:
\n\"Envision
\nWe know that,
\nThe initial value is also known as the y-intercept
\nThe y-intercept is the value of y when x = 0
\nSo,
\nFor Function A,
\nThe initial value (y-intercept) is: 4
\nFor function B,
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the rate of change
\nc is the initial value or the y-intercept
\nSo.
\nThe initial value (y-intercept) is: 3
\nHence, from the above,
\nWe can conclude that Function A has the greater initial value<\/p>\n

Question 8.
\nTell whether each function is linear or nonlinear.
\n\"Envision
\nAnswer:
\nThe given functions are:
\n\"Envision
\nWe know that,
\nTo find whether the given function is linear or non-linear, we have to verify whether the rate of change is constant or not
\nIf the rate of change is constant, then the function is linear
\nIf the rate of change is not constant, then the function is non-linear
\nNow,
\nFor Function A,
\nThe rate of change = \\(\\frac{y}{x}\\)
\nFor x = 1 and y = 2,
\nThe rate of change = 2
\nFor x = 2 and y = 5,
\nThe rate of change = 2.5
\nHence,
\nFunction A is a non-linear function
\nNow,
\nFor Function B,
\nThe rate of change = \\(\\frac{y}{x}\\)
\nFor x = 1 and y = 4,
\nThe rate of change = 4
\nFor x = 1.5 and y = 3,
\nThe rate of change = 2
\nHence,
\nFunction B is a non-linear function<\/p>\n

Question 9.
\nTell whether each function is linear or nonlinear.
\n\"Envision
\nAnswer:
\nThe given functions are:
\n\"Envision
\nWe know that,
\nFor a relation to be a graph, each input has a different output but the same input will not have different outputs
\nNow,
\nFrom Function A,
\nWe can observe that there are different inputs for different outputs i.e., the values of x and y are different
\nSo,
\nThe rate of change is not constant since the graph is non-linear
\nHence,
\nFunction A is a non-linear function
\nNow,
\nFrom Function B,
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm = 1
\nc = 0
\nSo,
\nThe rate of change is constant for Function B
\nHence,
\nFunction B is a linear function<\/p>\n

Question 10.<\/span><\/p>\n

Determine whether each function is linear or nonlinear from its graph.
\n\"Envision
\nAnswer:
\nThe given graphs are:
\n\"Envision
\nWe know that,
\nIf the graph is a straight line, then the function is a linear function
\nIf the graph is not a straight line, then the function is not a linear function
\nHence, from the above,
\nWe can conclude that
\nThe function 1 is a linear function
\nThe function 2 is a non-linear function<\/p>\n

Question 11.
\nLook for Relationships<\/strong> Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nJustin opens a savings account with $4. He saves $2 each week.
\nNow,
\nThe given table is:
\n\"Envision
\nFrom the given table,
\nWe can observe that there is a constant rate of change
\nNow,
\nFor weeks,
\nWe can obtain the number of weeks by adding 1 i.e., 0 + 1, 1 + 1, etc
\nSo,
\nThe rate of change is constant i.e., 1
\nFor money in account,
\nWe can obtain the money by adding 2 to the initial amount of money i.e., 4 + 2, 6 + 2, etc
\nSo,
\nThe rate of change is constant i.e., 2
\nHence, from the above,
\nWe can conclude that since the rate of change is constant, the given situation represents a linear function<\/p>\n

Question 12.
\nReasoning<\/strong> The function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores. What do the rates of change tell you about how each player earns points?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe function y = 4x + 3 describes Player A’s scores in a game of trivia, where x is the number of questions answered correctly and y is the score. The function represented in the table shows Player B’s scores.
\nNow,
\nFor Player A,
\nThe given equation is:
\ny = 4x + 3
\nCompare the givene quation with
\ny = mx + c
\nWhere,
\nm is the rate of change
\nc is the y-intercept
\nSo,
\nThe rate of change of Player A is: 4
\nNow,
\nThe given table is:
\n\"Envision
\nFor Player B,
\nFrom the given table,
\nWe can observe that the rate of change is constant for both the values of x and y
\nThe rate of change for both the values of x and y is: 1
\nSo,
\nThe rate of change of Player B is: 1
\nSo,
\nThe rate of change of Player A > The rate of change of Player B
\nHence, from the above,
\nWe can conclude that Player A earns more points<\/p>\n

Question 13.
\nTwo athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table. Compare the initial values for each. What does the initial value mean in this situation?
\n\"Envision
\nAnswer:
\nIt is given that
\nTwo athletes are training over a two-week period to increase the number of push-ups each can do consecutively. Athlete A can do 16 push-ups to start, and increases his total by 2 each day. Athlete B’s progress is charted in the table
\nNow,
\nFor Athlete A,
\nThe starting point is: 16
\nSo,
\nThe initial point for Athlete A is: 16
\nNow,
\nFor Athlete B,
\nThe given table is:
\n\"Envision
\nWe know that,
\nThe initial point or the y-intercept is the value of y when x = 0
\nSo,
\nThe initial point for Athlete B is: 12
\nHence, from the above,
\nWe can conclude that
\nThe initial points in the given situation describe the number of pushups one can do at a time without stop
\nThe initial point for Athlete A > The initial point for Athlete B<\/p>\n

Question 14.
\nHigher-Order Thinking<\/strong> The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions. Which function has the greatest rate of change? Which has the least? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions
\nNow,
\na)
\nThe given equation is:
\ny = 4x – 2
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the constant rate of change
\nSo,
\nFor the given equation,
\nThe rate of change is: 4
\nb)
\nThe given table and graph are:
\n\"Envision
\nNow,
\nFrom the given table,
\nThe rate of change = \\(\\frac{y}{x}\\)
\nFor x = 1 and y = 5,
\nThe rate of change is: 5
\nFor x = 2 and y = 10
\nThe rate of change is: 5
\nNow,
\nSince the rate of change is constant for all the cases,
\nThe rate of change for the given table is: 5
\nNow,
\nFrom the given graph,
\nThe given points to find the slope are: (0, 4), and (2, 0)
\nSo,
\nSlope (or) The rate of change = \\(\\frac{0 – 4}{2 – 0}\\)
\n= \\(\\frac{-4}{2}\\)
\n= -2
\nNow,
\nWhen we compare the rate of change for all the three linear functions,
\nThe rate of change of the table > The rate of change of the equation < The rate of change of the graph
\nHence, from the above,
\nWe can conclude that
\nThe function that has the greatest rate of change is: Table
\nThe function that has the least rate of change is: Graph<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 15.
\nThe students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time. Which snack did the students eat at a faster rate? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time.
\nNow,
\nThe rate of change of grapes consumption is: 12 grapes per minute
\nNow,
\nThe given table is:
\n\"Envision
\nNow,
\nThe rate of change of carrots consumption = \\(\\frac{The difference between any 2 values of carrots remaining}{The values of the tie elapsed corresponding to the carrots remaining}\\)
\n= \\(\\frac{118 – 136}{8 – 6}\\)
\n= –\\(\\frac{18}{2}\\)
\n= -9 carrots per minute
\nSo,
\nThe consumption rate of grapes > The consumption rate of carrots
\nHence, from the baove,
\nWe can conclude that grapes can be eaten at a faster rate<\/p>\n

Question 16.
\nThe height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours. Which of the following statements are true?
\n\"Envision The initial height of Candle A is greater than the initial height of Candle B.
\n\"Envision The height of Candle A decreases at a faster rate than the height of Candle B.
\n\"Envision Candle B will burn out in 58 hours.
\n\"Envision After 10 hours, the height of Candle A is 110 millimeters.
\n\"Envision Candle A will burn out before Candle B.
\nAnswer:
\nLet the given options be named as A, B, C, D, and E
\nIt is given that
\nThe height of a burning candle can be modeled by a linear function. Candle A has an initial height of 201 millimeters, and its height decreases to 177 millimeters after 4 hours of burning. The height, h, in millimeters, of Candle B, can be modeled by the function h = 290 – 5t, where t is the time in hours
\nNow,
\nThe rate of change of Candle A = \\(\\frac{201 – 177}{4}\\)
\n= \\(\\frac{24}{4}\\)
\n= 6 millimeters per hour
\nNow,
\nFor Candle B,
\nThe time to burn the Candle B = \\(\\frac{290}{5}\\)
\n= 58 hours
\nHence, from the above,
\nWe can conclude that B, C, and E match with the given situation<\/p>\n

Topic 3 MID-TOPIC CHECKPOINT<\/h3>\n

Question 1.
\nVocabulary<\/strong> How can you determine whether a relation is a function? Lesson 3.1
\nAnswer:
\nIdentify the output values.\u00a0If\u00a0each input value leads to only one output value, classify the\u00a0relationship\u00a0as a\u00a0function.\u00a0If\u00a0any input value leads to two or more outputs, do not classify the\u00a0relationship\u00a0as a\u00a0function.<\/p>\n

Question 2.
\nCan an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? Explain your answer. Lesson 3.1
\n\"Envision
\nAnswer:
\nThe given diagram is:
\n\"Envision
\nFor the relation in the diagram to be a function,
\nThe arrows from the right side to the left side can be many but the arrows from the left side to the right side can only be one<\/p>\n

Question 3.
\nTwo linear functions are shown below. Which function has the greater rate of change? Justify your response. Lesson 3.3
\n\"Envision
\nAnswer:
\nThe given linear functions are:
\n\"Envision
\nWe know that,
\nFor a linear function,
\nThe rate of change is constant
\nSo,
\nFor Function A,
\nThe rate of change = \\(\\frac{y}{x}\\)
\n= \\(\\frac{3}{6}\\)
\n= \\(\\frac{1}{2}\\)
\n= 0.5
\nFor Function B,
\nThe given equation is:
\ny = \\(\\frac{1}{2}\\)x – 1
\ny = 0.5x – 1
\nCompare the above equation with
\ny = mx + c
\nWhere,
\nm is the rate of change
\nSo,
\nThe rate of change is: 0.5
\nHence, from the above,
\nWe can conclude that the two linear functions have the same rate of change<\/p>\n

Question 4.
\nNeil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table. Is the relation of hours of study time to the grade earned on a test a function? Explain why. Use the graph to justify your answer. Lesson 3.2
\n\"Envision
\nAnswer:
\nIt is given that
\nNeil took 3 math tests this year. The number of hours he spent studying for each test and the corresponding grades he earned is shown in the table.
\nNow,
\nFor the number of hours as input and the Grades as the output,
\nWe can observe that
\nFor each value of the hours, there are the same values of the Grade
\nSo,
\nThe relation of hours of study time to the grade earned on a test is not considered a function
\nNow,
\nThe representation of the relation in the form of the ordered pairs is:
\n(4, 75), (6, 75), and (6, 82)
\nHence,
\nThe representation of the ordered pairs in the coordinate plane is:
\n\"\"<\/p>\n

Question 5.
\nIs the function shown linear or nonlinear? Explain your answer. Lesson 3.3
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nFor each value of x, there is only 1 value of y
\nSo,
\nThe given graph is a function
\nWe know that,
\nA function is called a linear if the graph is a straight line
\nA function is called non-linear if the graph is in any shape other than the straight line
\nHence, from the above,
\nWe can conclude that the given graph is a non-linear function<\/p>\n

How well did you do on the mid-topic checkpoint? Fill in the stars.\"Envision<\/p>\n

Topic 3 MID-TOPIC PERFORMANCE TASK<\/h3>\n

Sarah, Gene, and Paul are proposing plans for a class fundraiser. Each presents his or her proposal for the amount of money raised, y, for x number of hours worked, in different ways.
\n\"Envision
\nAnswer:
\nThe given graphs are:
\n\"Envision
\nNow,
\na)
\nFrom the given graph,
\nWe can observe that
\nFor each value of x, there is only 1 value of y
\nThe given graph is a straight line
\nNow,
\nThe rate of change of the given graph = \\(\\frac{y}{x}\\)
\n= \\(\\frac{0 – 10}{1 – 0}\\)
\n= -10
\nHence,
\nThe given graph is considered a linear function
\nb)
\nFrom the given table,
\nTo consider a function linear, verify whether the rate of change is constant or not
\nNow,
\nFor all the values of x,
\nThe rate of change is: 5
\nFor all the values of y,
\nThe rate of change is: 35
\nSo,
\nThe rate of change of the given table = \\(\\frac{y}{x}\\)
\n= \\(\\frac{5}{5}\\)
\n= 7
\nHence,
\nThe given table is considered a linear function
\nc)
\nThe given equation is:
\ny = 10x + 7
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the rate of change
\nSo,
\nThe rate of change for the given equation is: 10
\nHence,
\nThe given equation is considered a linear function<\/p>\n

PART A<\/strong>
\nIs each of the proposals represented by linear functions? Explain.
\nAnswer:
\nYes, all the proposals are represented by linear functions<\/p>\n

PART B<\/strong>
\nDoes the class have any money in the account now? How can you tell?
\nAnswer:
\nFrom the given graph,
\nWe can observe that the straight line does not start from 0 but from 10
\nSo,
\nThe initial value of the graph will be: 10
\nWe know that,
\nThe initial value is considered the y-intercept
\nHence, from the above,
\nWe can conclude that the class have the money in the account now i.e., $10<\/p>\n

PART C<\/strong>
\nWhich fundraising proposal raises money at the fastest rate? Explain.
\nAnswer:
\nSince the rate of change is the highest for Paul’s proposal,
\nPaul’s Proposal raises money at the fastest rate<\/p>\n

PART D<\/strong>
\nIf Sarah and her classmates are hoping to raise $200, which proposal do you recommend that Sarah and her classmates choose? Explain why you recommend that proposal.
\nAnswer:
\nIt is given that Sarah and her classmates are hoping to raise $200
\nSo,
\nTo raise the money,
\nWe have to choose the plan which has the highest rate of change
\nHence, from the above,
\nWe can conclude that Sarah and her classmates choose Paul’s proposal<\/p>\n

3-Act Mathematical Modeling: Every Drop Counts<\/h3>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision<\/p>\n

\u0410\u0421\u0422 1<\/strong><\/p>\n

Question 1.
\nAfter watching the video, what is the first question that comes to mind?
\nAnswer:
\nAfter watching the video,
\nThe first question that comes to mind is:
\nHow much amount of water people waste brushing their teeth?<\/p>\n

Question 2.
\nWrite the Main Question you will answer.
\nAnswer:
\nThe main question you will answer is:
\nHow much amount of water people waste brushing their teeth?<\/p>\n

Question 3.
\nConstruct Arguments<\/strong> Predict an answer to this Main Question. Explain your prediction.
\n\"Envision
\nAnswer:
\nThe answer to the main question is: 4 gallons
\nThe prediction of the answer for the main question is according to the surveys done by International Organisations<\/p>\n

Question 4.
\nOn the number line below, write a number that is too small to be the answer. Write a number that is too large.
\n\"Envision
\nAnswer:
\nFrom the above,
\nWe can observe that
\nThe maximum amount (Too large) of water used to brush teeth is: 4 gallons
\nThe minimum amount (Too small) of water used to brush the teeth is: 2 gallons
\nHence,
\nThe representation of the amounts of water used to brush teeth in this situation is:
\n\"\"<\/p>\n

Question 5.
\nPlot your prediction on the same number line.
\nAnswer:
\nFrom the above,
\nWe can observe that there are minimum and maximum amounts of water used to brush the teeth<\/p>\n

Now,
\nLet x be the amount of water used to brush the teeth
\nSo,
\nThe prediction will be: 2 < x < 4
\nHence,
\nThe representation of the prediction on the number line is:
\n\"\"<\/p>\n

ACT 2<\/strong><\/p>\n

Question 6.
\nWhat information in this situation would be helpful to know? How would you use that information?
\n\"Envision
\nAnswer:
\nThe information in this situation that would be helpful to know is:
\nHow much time did it take to completely brush your teeth?
\nFrom the above information,
\nWe can estimate the amount of water used to brush your teeth<\/p>\n

Question 7.
\nUse Appropriate Tools<\/strong> What tools can you use to solve the problem? Explain how you would use them strategically.
\nAnswer:<\/p>\n

Question 8.
\nModel with Math<\/strong>
\nRepresent the situation using mathematics.
\nUse your representation to answer the Main Question.
\nAnswer:<\/p>\n

Question 9.
\nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
\n\"Envision
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Question 10.
\nWrite the answer you saw in the video.
\nAnswer:<\/p>\n

Question 11.
\nReasoning<\/strong> Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
\n\"Envision
\nAnswer:<\/p>\n

Question 12.
\nMake Sense and Persevere<\/strong> Would you change your model now that you know the answer? Explain.
\nAnswer:<\/p>\n

ACT 3<\/strong><\/p>\n

Reflect<\/strong><\/p>\n

Question 13.
\nModel with Math<\/strong>
\nExplain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
\n\"Envision
\nAnswer:<\/p>\n

Question 14.
\nBe Precise<\/strong> How do the units you chose and the method you used help you communicate your answer?
\nAnswer;<\/p>\n

SEQUEL<\/strong><\/p>\n

Question 15.
\nUse Structure<\/strong> How much water will he save in a year?
\n\"Envision
\nAnswer:<\/p>\n

Lesson 3.4 Construct Functions to Model Linear Relationships<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explore It!<\/strong>
\nErick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.
\n\"Envision
\nAnswer:
\nIt is given that
\nErick wants to buy a new mountain bike that costs $250. He has already saved $120 and plans to save $20 each week from the money he earns for mowing lawns. He thinks he will have saved enough money after seven weeks.
\nSo,
\nThe total amount he saved = The amount he saved already + The amount he planned to save each week
\nLet,
\nThe number of weeks —-> x
\nThe total amount he saved —-> y
\nSo,
\ny = $20x + $120
\nCompare the above equation with y = mx + c
\nWhere,
\nm s the rate of change (or) slope
\nc is the y-intercept
\nSo,
\nFor the given equation,
\nThe rate of change (m) is: 20
\nThe y-intercept is: 120<\/p>\n

A. Complete the table. Then graph the data.
\n\"Envision
\nAnswer:
\nThe equation is:
\ny = $20x + $120
\nSo,
\nThe completed table for the above equation is:
\n\"\"
\nSo,
\nThe representation of the given equation in the coordinate plane is:
\n\"\"<\/p>\n

B. How can you tell that the relationship is a linear function from the table? How can you tell from the graph?
\nAnswer:
\nWe know that,
\nA relation is said to be a function when an input value matches only with an output value
\nA function is said to be a linear function when the graph of that function is a straight line
\nSo,
\nFrom part (a),
\nFrom the table,
\nWe can observe that for each value of the week, there is a different amount saved
\nSo,
\nFrom the table,
\nThe given relation is said to be a function
\nFrom the graph of the equation,
\nWe can observe that the graph is a straight line
\nSo,
\nWe can say that the function is a linear function<\/p>\n

Focus on math practices<\/strong>
\nGeneralize How can the different representations help you determine the properties of functions?
\nAnswer:
\nThe different representations of the functions are:
\nA) Symbolic or Algebraic representation – The properties can be found out by comparing with the standard form
\nB) Numerical (Tables) representation – The properties can be found out by the rate of change and the initial values
\nC) Graphical representation – The properties can be found out by the values of x and y
\nD) Verbal representation – The properties can be found out by the keywords<\/p>\n

? Essential Question<\/strong>
\nHow can you use a function to represent a linear relationship?
\nAnswer:
\nAnother approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.<\/p>\n

Try It!<\/strong>
\nHow will the height of the ramp change if the plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long?
\n\"Envision
\nGraph the function. The slope of the function shown in the graph is \"Envision. The equation of the function is y = \"Envision
\nx. If the base length is 110 inches, then the height of the ramp will be \"Envision inches.
\nAnswer:
\nIt is given that
\nThe plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long
\nSo,
\nThe rate of change (m) = \\(\\frac{Rise}{Run}\\)
\nm = \\(\\frac{3}{15}\\)
\nm = \\(\\frac{1}{5}\\)
\nSo,
\nThe slope of the function shown in the graph is: \\(\\frac{1}{5}\\)
\nNow,
\nWe know that,
\nThe representation of the linear equation is:
\ny = mx
\nSo,
\ny = \\(\\frac{1}{5}\\)x
\nWhere,
\ny is the height of the ramp
\nx is the base length of the triangle
\nNow,
\nFor the base length of 110 inches,
\ny = \\(\\frac{1}{5}\\) (110)
\ny = 22 inches
\nHence, from the above,
\nWe can conclude that the height of the ramp will be 22 inches for the base length of 110 inches<\/p>\n

Convince Me!<\/strong>
\nExplain why the initial value and the y-intercept are equivalent.
\nAnswer:
\nAn equation in slope-intercept\u00a0form of a line includes the slope and the\u00a0initial value\u00a0of the function. The\u00a0initial value, or\u00a0y-intercept, is the output\u00a0value\u00a0when the input of a linear function is zero. It is the\u00a0y-value\u00a0of the point where the line crosses the\u00a0y-axis.<\/p>\n

Try It!<\/strong>
\nJin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\\(\\frac{1}{2}\\) cups of dog food. After 5 weeks, he has used 21\\(\\frac{1}{4}\\) cups. Construct a function in the form y = mx + b to represent the amount of dog food used, y, after x weeks.
\nAnswer:
\nIt is given that
\nJin is tracking how much food he feeds his dogs each week. After 2 weeks, he has used 8\\(\\frac{1}{2}\\) cups of dog food. After 5 weeks, he has used 21\\(\\frac{1}{4}\\) cups.
\nNow,
\nLet x be the number of weeks
\nLet y be the number of cups of dog food used
\nSo,
\nFor x = 2, y = 8.5 (The value of 8\\(\\frac{1}{2}\\))
\nFor x = 5, y = 21.25 (The value of 21\\(\\frac{1}{4}\\))
\nWe know that,
\nThe equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\n8.5 = 2m + c —–> (1)
\n21.25 = 5m + c ——> (2)
\nSolve eq (1) and eq (2)
\nSo,
\n8.5 = 2m + 21.25 – 5m
\n8.5 – 21.25 = 2m – 5m
\n-12.75 = -3m
\n3m = 12.75
\nm = \\(\\frac{12.75}{3}\\)
\nm = 4.25
\nNow,
\nFor the value of c,
\nSubstitute the value of m either in eq (1) or in eq (2)
\nSo,
\n8.5 = 2m + c
\n8.5 = 2 (4.25) + c
\n8.5 – 8.5 = c
\nc = 0
\nHence, from the above,
\nWe can conclude that the representation of the amount of dog used y, after x weeks in the equation form is:
\ny = 4.25x<\/p>\n

Try It!<\/strong>
\nThe graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing. What function in the form y = mx + b represents this relationship?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that there is an initial value or y-intercept
\nSo,
\nThe y-intercept from the given graph is: 1
\nNow,
\nTo find the slope from the given graph,
\nThe given points are: (30, 4), and (10, 2)
\nSo,
\nSlope (m) = \\(\\frac{2 – 4}{10 – 30}\\)
\nm = \\(\\frac{-2}{-20}\\)
\nm = \\(\\frac{1}{10}\\)
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = \\(\\frac{1}{10}\\)x + 1
\nHence, from the above,
\nWe can conclude that the equation that represents the given situation is:
\ny = \\(\\frac{1}{10}\\)x + 1<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

A function in the form y= mx + b represents a linear relationship between two quantities, x, and y.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n?Essential Question<\/strong> How can you use a function to represent a linear relationship?
\nAnswer:
\nAnother approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and c is the initial value of the dependent variable.<\/p>\n

Question 2.
\nMake Sense and Persevere<\/strong> Tonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10). What must the value of x be? Explain.
\nAnswer:
\nIt is given that
\nTonya is looking at a graph that shows a line drawn between two points with a slope of -5. One of the points is smudged and she cannot read it. The points as far as she can tell are (3, 5) and (x, 10)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1<\/sub>
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\n–5 = \\(\\frac{10 – 5}{x – 3}\\)
\n-5 = \\(\\frac{5}{x – 3}\\)
\n-5 (x – 3) = 5
\n-5 (x) + 5 (3) = 5
\n-5x + 15 = 5
\n-5x = 5 – 15
\n-5x = -10
\n5x = 10
\nx = \\(\\frac{10}{5}\\)
\nx = 2
\nHence, from the above,
\nWe can conclude that the value of x is: 2<\/p>\n

Question 3.
\nReasoning<\/strong> What is the initial value of all linear functions that show a proportional relationship?
\nAnswer:
\nWe know that,
\nThe representation of the proportional relationship is:
\ny = mx + 0
\nWhere,
\nm is the slope (or) rate of change
\nSo,
\nFrom the above equation,
\nWe can say that the initial value (or) the y-intercept is: 0
\nHence, from the above,
\nWe can conclude that the initial value of all linear functions that show a proportional relationship is: 0<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWrite a function in the form y = mx + b for the line that contains the points (-8.3, -5.2) and (6.4, 9.5).
\nAnswer:
\nThe given points are:
\n(-8.3, -5.2) and (6.4, 9.5)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{9.5 + 5.2}{6.4 + 8.3}\\)
\nm = \\(\\frac{14.7}{14.7}\\)
\nm = 1
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n-5.2 = -8.3 + c
\nc = 8.3 – 5.2
\nc = 3.1
\nHence, from the above,
\nWe can conclude that the representation of the linear equation for the given points is:
\ny = x + 3.1<\/p>\n

Question 5.
\nThe data in the table below represent a linear relationship. Fill in the missing data.
\n\"Envision
\nAnswer:
\nIt is given that the data in the table represent a linear relationship
\nSo,
\nFor a linear relationship, for each value of x, there is only 1 value of y
\nNow,
\nTo find the missing data,
\nFind the rate of change for x and y
\nFor all the values of x,
\nThe rate of change = 20 – 10
\n= 10
\nFor all the values of y,
\nThe rate of change = 15 – 10
\n= 5
\nHence,
\nThe completed table with the missing data is:
\n\"\"<\/p>\n

Question 6.
\nWhat is an equation that represents the linear function described by the data in Item 5?
\nAnswer:
\nFrom the data in Item 5,
\nSlope (m) = \\(\\frac{The rate of change of y}{The rate of change of x}\\)
\nm = \\(\\frac{5}{10}\\)
\nm = \\(\\frac{1}{2}\\)
\nWe know that,
\nThe equation that represents a linear relationship is:
\ny = mx
\nSo,
\ny = \\(\\frac{1}{2}\\)x
\nHence, from the above,
\nThe equation that represents the linear function described by the data in Item 5 is:
\ny = \\(\\frac{1}{2}\\)x<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 7.
\nA line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.
\nAnswer:
\nThe given points are:
\n(4, 19) and (9, 24)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{24 – 19}{9 – 4}\\)
\nm = \\(\\frac{5}{5}\\)
\nm = 1
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n19 = 4 + c
\nc = 19 – 4
\nc = 15
\nHence, from the above,
\nWe can conclude that the representation of the linear equation for the given points is:
\ny = x + 15<\/p>\n

Question 8.
\nWhat is a linear function in the form y = mx + b for the line passing through (4.5, -4.25) with y-intercept 2.5?
\nAnswer:
\nIt is given that
\nA line passing through (4.5, -4.25) with y-intercept 2.5
\nWe know that,
\nThe y-intercept is the value of y when x = 0
\nNow,
\nThe given points are:
\n(4.5, -4.25) and (0, 2.5)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{2.5 + 4.25}{0 – 4.5}\\)
\nm = \\(\\frac{6.75}{-4.5}\\)
\nm = -1.5
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = -1.5x + 2.5
\nHence, from the above,
\nWe can conclude that the representation of the linear equation for the given points is:
\ny = -1.5x + 2.5<\/p>\n

Question 9.
\nA car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet. What linear function in the form y = mx + b represents the distance in feet, d, the car has traveled any number of seconds, t, after passing the timing device?
\nAnswer:
\nIt is given that
\nA car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nSo,
\nFor a constant speed,
\nTime is considered as input and distance is considered as the output
\nIt is also given that time will start from (0, 0) i..e, at t = 0
\nSo,
\nThe representation of the equation that passes through the origin is:
\ny = mx
\nWhere,
\nm is the rate of change (or) slope
\nIn this situation,
\nThe rate of change = \\(\\frac{Distance}{Time}\\)
\nSo,
\ny = \\(\\frac{840}{8}\\)x
\ny = 105x
\nHence, from the above,
\nWe can conclude that the linear equation that represents the distance traveled by a car after passing the timing device is:
\ny = 105x<\/p>\n

Question 10.
\nAt time t = 0, water begins to drip out of a pipe into an empty bucket. After 56 minutes, 8 inches of water are in the bucket. What linear function in the form y = mx + b represents the amount of water in inches, w, in the bucket after t minutes?
\nAnswer:
\nIt is given that
\nAt time t = 0, water begins to drip out of a pipe into an empty bucket
\nNow,
\nIn this situation,
\nTime is considered as output and the amount of water in the bucket is considered as the input
\nIt is also given that time will start from (0, 0) i..e, at t = 0
\nSo,
\nThe representation of the equation that passes through the origin is:
\ny = mx
\nWhere,
\nm is the rate of change (or) slope
\nIn this situation,
\nThe rate of change = \\(\\frac{Time}{The amount of water}\\)
\nSo,
\ny = \\(\\frac{56}{8}\\)x
\ny = 7x
\nHence, from the above,
\nWe can conclude that the linear equation that represents the amount of water in inches, w, in the bucket after t minutes
\ny = 7x<\/p>\n

Question 11.
\nThe graph of the line represents the cost of renting a kayak. Write a linear function in the form y = mx + b to represent the relationship of the total cost, c, of renting a kayak for t hours.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph of the line represents the cost of renting a kayak
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that the line does not pass through the origin and it has the initial value i..e, the y-intercept
\nSo,
\nFrom the graph,
\nThe y-intercept is: 8
\nNow,
\nTo find the slope,
\nThe points from the graph are: (2, 12), and (4, 16)
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{16 – 12}{4 – 2}\\)
\nm = \\(\\frac{4}{2}\\)
\nm = 2
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = 2x + 8
\nHence, from the above,
\nWe can conclude that the relationship to represent the total cost, c, of renting a kayak for t hours is:
\ny = 2x + 8<\/p>\n

Question 12.
\nAn online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
\na. Write a linear function in the form y = mx + b that represents the total cost, y, in dollars, for a single order of x sweatshirts.
\nAnswer:
\nIt is given that
\nAn online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
\nSo,
\nThe total cost of the sweatshirts = The cost of each sweatshirt + The flat fee of the sweatshirt for shipping
\nLet the number of sweatshirts be x
\nlet the total cost of the sweatshirts be y
\nSo,
\ny = $3.99 + $6.50x
\nNow,
\nWe know that,
\nThe linear equation representation in the slope-intercept form is:
\ny = mx + c
\nHence, from the above,
\nWe can conclude that the equation that represents the total cost, y, in dollars, for a single order of x sweatshirts is:
\ny = $3.99 + 6.50x<\/p>\n

b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.
\nAnswer:
\nFrom part (a),
\nWe know that,
\nThe equation that represents the total cost, y, in dollars, for a single order of x sweatshirts, is:
\ny = $3.99 + $6.50x
\nWhere,
\n$3.99 is the shipping charge for x sweatshirts,
\nNow,
\nIf the shipping charge applied to each sweatshirt, then
\nThe representation of the linear equation which we obtained in part (a) is:
\ny = $6.50x + \\(\\frac{$3.99}{x}\\)
\nHence, from the above,
\nWe can conclude that the linear function that represents if the shipping charge applied to each sweatshirt is:
\ny = $6.50x + \\(\\frac{$3.99}{x}\\)<\/p>\n

Question 13.
\nA store sells packages of comic books with a poster.
\na.<\/strong> Model with Math<\/strong> Write a linear function in the form y = mx + b that represents the cost, y, of a package containing any number of comic books, x.
\nb. Construct Arguments<\/strong> Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.
\n\"Envision
\nAnswer:
\na)
\nIt is given that a store seller sells packages of comic books with a poster
\nNow,
\nLet x be the number of comics
\nLet y be the amount obtained by selling comics & poster
\nSo,
\nFor x = 6, y = $12.75
\nFor x = 13, y = $19.75
\nWe know that,
\nThe representation of the linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nTo find the slope,
\nThe points are: (6, 12.75), and (13, 19.75)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{19.75 – 12.75}{13 – 6}\\)
\nm = \\(\\frac{7}{7}\\)
\nm = 1
\nSo,
\ny = x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n12.75 = 6 + c
\nc = 12.75 – 6
\nc = 6.75
\nHence, from the above,
\nWe can conclude that the representation of the linear equation that represents the cost, y, of a package containing any number of comic books, x is:
\ny = x + 6.75
\nb)
\nIt is given that another store sells a similar package, modeled by a linear function with an initial value of $7.99
\nNow,
\nFrom part (a),
\nThe initial value is: $6.75
\nFrom the above initial values,
\nWe can observe that
\n$6.75 < $7.99
\nHence, from the above,
\nWe can conclude that another store has the better deal<\/p>\n

Question 14.
\nHigher-Order Thinking<\/strong> Recommendations for safely thawing a frozen turkey are provided on the packaging.
\na. What is the thaw rate of the turkey in hours per pound for refrigerator thawing? For cold water thawing?
\nb. Write a linear function in the form y = mx + b to represent the time, t, in hours it takes to thaw a turkey in the refrigerator as a function of the weight, w, in pounds of the turkey.
\n\"Envision
\nAnswer:
\na)
\nWe know that,
\nThe thaw rate is nothing but the rate of change
\nSo,
\nFor refrigerator thawing,
\nRate of change = \\(\\frac{The number of pounds}{Time}\\)
\n= 4 pounds per day
\nWe know that
\n1 day = 24 hours
\nSo,
\nRate of change = \\(\\frac{4}{24}\\)
\n= \\(\\frac{1}{6}\\) pound per hour
\nFor Cold water thawing,
\nRate of change = \\(\\frac{The number of pounds}{Time}\\)
\n= 1 pound per 30 minutes
\nWe know that
\n1 hour = 60minutes
\nSo,
\nRate of change = \\(\\frac{1 (2)}{1}\\)
\n= 2 pounds per hour
\nHence, from the above,
\nWe can conclude that
\nThe rate of change for refrigerator thawing is: \\(\\frac{1}{6}\\) pounds per hour
\nThe rate of change for cold water thawing is: 2 pounds per hour
\nb)
\nWe know that,
\nThe representation of the linear equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\nFor refrigerator thawing,
\nThe representation of the linear equation as a function of weight w in pounds is:
\ny = \\(\\frac{1}{6}\\)x + c
\nFor the value of c,
\nSubstitute (24, 4) in the above equation [ The time as x in hours and the weight as y]
\nSo,
\n4 = \\(\\frac{24}{6}\\) + c
\n4 = 4 + c
\nc = 0
\nHence, from the above,
\nWe can conclude that the representation of the linear equation as a function of weight w in pounds is:
\ny = \\(\\frac{1}{6}\\)x<\/p>\n

Question 15.
\nReasoning<\/strong> The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
\na. What is the constant rate of change? What does it represent?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nTo find the rate of change,
\nThe points are: (20, 450), and (10, 300)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{300 – 450}{10 – 20}\\)
\nm = \\(\\frac{150}{10}\\)
\nm = 15
\nHence, from the above,
\nWe can conclude that the constant rate of change for the given graph is: 15<\/p>\n

b. What is the initial value? What might that represent?
\nAnswer:
\nWe know that,
\nThe initial value is nothing but the y-intercept
\nHence, from the above,
\nWe can conclude that the initial value is 50 and this value represents the initial cost of the mulch<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 16.
\nAn international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75. Which linear function represents the cost, y, for any number of samples, x?
\nA. y = x + 2.75
\nB. y = 3x + 2.75
\nC. y = x + 3
\nD. y = 3x + 3
\nAnswer:
\nIt is given that
\nAn international food festival charges for admission and for each sample of food. Admission and 3 samples cost $5.75. Admission and 6 samples cost $8.75
\nNow,
\nLet x be the number of samples
\nLet y be the cost of samples &Admission
\nSo,
\nFor x = 3, y = $5.75
\nFor x = 6, y = $8.75
\nWe know that,
\nThe representation of the linear equation in the slope-intercept form is:
\ny = mx + c
\nWhere,
\nm is the slope
\nc is the y-intercept
\nNow,
\nTo find the slope,
\nThe points are: (3, 5.75), and (6, 8.75)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{8.75 – 5.75}{6 – 3}\\)
\nm = \\(\\frac{3}{3}\\)
\nm = 1
\nSo,
\ny = x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n5.75 = 3 + c
\nc = 5.75 – 3
\nc = 2.75
\nSo,
\ny = x + 2.75
\nHence, from the above,
\nWe can conclude that option A matches the given situation<\/p>\n

Question 17.
\nSome eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.<\/p>\n

PART A<\/strong>
\nWrite a linear function in the form y = mx + b that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x.
\nAnswer:
\nIt is given that
\nSome eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.
\nNow,
\nLet x be the number of hours
\nSo,
\nThe total number of muffins students will make = The number of muffins that the students had already made + The number of muffins the students will make in x hours if they made 40 muffins in an hour
\ny = 40x + 200
\nWe know that,
\nThe representation of the linear equation in the slope-intercept form is:
\ny = mx + c
\nHence, from the above,
\nWe can conclude that
\nA linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
\ny = 40x + 200<\/p>\n

PART B<\/strong>
\nHow many additional hours would the students spend to make 640 muffins?
\nAnswer:
\nFrom part (a),
\nA linear function that represents the total number of muffins the students will make, y, and the number of additional hours spent making the muffins, x is:
\ny = 40x + 200
\nIt is given that the number of muffins students made is: 640
\nSo,
\ny = 640
\nSo,
\n640 = 40x + 200
\n40x = 640 – 200
\n40x = 440
\n4x = 44
\nx = \\(\\frac{44}{4}\\)
\nx = 11 hours
\nHence, from the above,
\nWe can conclude that the additional hours the students would spend to make 640 muffins is: 11 hours<\/p>\n

Lesson 3.5 Intervals of Increase and Decrease<\/h3>\n

Solve & Discuss It!<\/strong><\/p>\n

Martin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill. How do you think the routes will differ? What do you think about the relationship between speed and time?
\n\"Envision
\nAnswer:
\nIt is given that
\nMartin will ride his bike from his house to his aunt’s house. He has two different routes he can take. One route goes up and down a hill. The other route avoids the hill by going around the edge of the hill.
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nIn this situation,
\nThe distance is constant for the 2 routes
\nSo,
\nAs speed increases, the time decreases
\nNow,
\nFor route 1:
\nRoute 1 is divided into 2 parts
\nFor the first part (Going up),
\nAs there is friction,
\nThe speed decreases as time increases
\nFor the second part (Going down),
\nThe speed increases as time decreases
\nFor route 2:
\nRoute 2 is divided into 2 parts
\nFor the first part (Going down),
\nThe speed increases as time decreases
\nFor the second part,
\nThe speed is constant as time increases<\/p>\n

Focus on math practices<\/strong>
\nReasoning How do the characteristics of each route affect Martin’s travel time and speed?
\nAnswer:
\nWe know that,
\nFor the constant distance,
\nSpeed \u221d \\(\\frac{1}{Time}\\)
\nNow,
\nRoute 1 is in the shape of a parabola
\nIn route 1,
\nFor the first part,
\nThe speed increases as time decreases
\nFor the second part,
\nThe speed decreases as time increases
\nNow,
\nRoute 2 is increasing for some time and later becomes constant
\nIn route 2,
\nFor the first part,
\nThe speed increases as time decreases
\nFor the second part,
\nThe speed is constant as time increases<\/p>\n

? Essential Question<\/strong>
\nHow does a qualitative graph describe the relationship between quantities?
\nAnswer:
\nThe formal term to\u00a0describe a straight-line\u00a0graph\u00a0is linear, whether or not it goes through the origin, and the\u00a0relationship\u00a0between the two variables is called a linear\u00a0relationship. Similarly, the\u00a0relationship\u00a0shown by a curved\u00a0graph\u00a0is called non-linear.<\/p>\n

Try It!<\/strong>
\nThe graph at the right shows another interval in the train’s travel. Which best describes the behavior of the train in the interval shown?
\nAs time \"Envision, the speed of the train
\n\"Envision
\nThe function is \"Envision
\nAnswer:
\nIt is given that the graph shows another interval in the train’s travel
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nAs speed decreases, the time increases
\nHence,
\nThe function of the given graph is decreasing in nature<\/p>\n

Convince Me!<\/strong>
\nHow would the graph of the function change if the speed of the train was increasing?
\nAnswer:
\nWe know that,
\nIf speed increases, then the time decreases for a constant distance
\nSo,
\nFor a constant distance,
\nSpeed \u221d \\(\\frac{1}{Time}\\)
\nHence,
\nThe graph of the function is decreasing in nature as the speed of the train is increased<\/p>\n

Try It!<\/strong>
\nWrite a scenario that the graph above could represent. (Example 3)
\nAnswer:
\nThe scenario that the given graph could represent is:
\nThe traveling of a vehicle on a hill<\/p>\n

KEY CONCEPT<\/strong>
\nYou can describe the relationship between two quantities by analyzing the behavior of the function relating the quantities in different intervals on a graph.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\nEssential Question<\/strong> How does a qualitative graph describe the relationship between quantities?
\nAnswer:
\nThe formal term to\u00a0describe a straight-line\u00a0graph\u00a0is linear, whether or not it goes through the origin, and the\u00a0relationship\u00a0between the two variables is called a linear\u00a0relationship. Similarly, the\u00a0relationship\u00a0shown by a curved\u00a0graph\u00a0is called non-linear.<\/p>\n

Question 2.
\nLook for Relationships<\/strong> How would knowing the slope of a linear function help determine whether a function is increasing or decreasing?
\nAnswer:
\nThe graph of an\u00a0increasing function has a positive slope. A line with a positive slope slants upward from left to right. For a decreasing function, the\u00a0slope\u00a0is negative. The output values\u00a0decrease\u00a0as the input values\u00a0increase.<\/p>\n

Question 3.
\nUse Structure<\/strong> What kind of graph of a function shows the same output values, or y-values, for each input value, or x-value?
\nAnswer:
\nThe vertical line test can be used to determine whether a\u00a0graph\u00a0represents a\u00a0function. A vertical line includes\u00a0all\u00a0points with a particular\u00a0x value. The\u00a0y value of\u00a0a point where a vertical line intersects a\u00a0graph\u00a0represents an\u00a0output\u00a0for that\u00a0input x value.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nWhat does the graph of the function at each interval represent?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nThe given graph is divided into 2 intervals
\nIn the first interval,
\nThe height increases with the increase of time
\nIn the second interval,
\nThe height decreases with the increase of time<\/p>\n

Question 5.
\nIn which intervals is the function increasing, decreasing, or constant?
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nHence, from the above,
\nWe can conclude that
\nThe intervals that the function is constant – 1, 5
\nThe intervals that the function is increasing – 3, 4
\nThe intervals that the function is decreasing – 2, 6<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nUse the graph to complete the statements.
\n\"Envision
\nThe function is \"Envision in intervals 1, 3, and 6.
\nThe function is \"Envision in intervals 2 and 5.
\nThe function is constant in interval \"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nHence, from the above,
\nWe can conclude that
\nThe intervals that the function is constant – 4
\nThe intervals that the function is increasing – 1, 3, 6
\nThe intervals that the function is decreasing – 2, 5<\/p>\n

Question 7.
\nThe graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner. Describe the relationship between the two quantities.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph below shows the temperature in Paula’s house over time after her mother turned on the air conditioner
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nAs time increases, the temperature decreases
\nHence, from the above,
\nWe can conclude that
\nTemperature (\u00b0F) \u221d \\(\\frac{1}{Time}\\)<\/p>\n

Question 8.
\nYou have a device that monitors the voltage across a lamp over time. The results are shown in the graph. Describe the behavior of the function in each interval.
\nIn interval (a), the function is \"Envision
\n\"Envision
\nIn the interval (b), the function is \"Envision
\nIn the interval (c), the function is \"Envision
\nIn interval (d), the function is \"Envision
\nAnswer:
\nIt is given that
\nYou have a device that monitors the voltage across a lamp over time. The results are shown in the graph
\nNow,
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nHence, from the above,
\nWe can conclude that
\nIn interval (a), the function is increasing
\nIn the Interval (b), the function is constant
\nIn the interval (c), the function is decreasing
\nIn the interval (d), the function is constant<\/p>\n

Question 9.
\nThe graph below shows the height of a roller coaster over time during a single ride. Circle the intervals in which the function is increasing. In which interval is the increase the greatest?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph below shows the height of a roller coaster over time during a single ride.
\nNow,
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nNow,
\nThe given graph with the intervals is:
\n\"\"
\nSo,
\nFrom the given graph,
\nThe intervals which are increasing in nature are: 1, 3, 5
\nHence, from the above,
\nWe can conclude that the increase is the greatest in the 1st interval<\/p>\n

Question 10.
\nReasoning<\/strong> The graph shows the speed of a car over time. What might the constant intervals in the function represent?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the speed of a car over time
\nNow,
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nWe know that,
\nSpeed = \\(\\frac{Distance}{Time}\\)
\nNow,
\nFrom the given graph,
\nWe can observe that
\nThe constant lines represent that the speed becomes constant even though the time is increasing
\nHence, from the above,
\nWe can conclude that the constant lines in the given graph represent the constant speed<\/p>\n

Question 11.
\nHigher-Order Thinking<\/strong> A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
\na. In how many intervals is the function decreasing?
\nb. How are the decreasing intervals alike?
\n\"Envision
\nAnswer:
\nIt is given that
\nA signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
\nNow,
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nNow,
\nThe given graph with the marked intervals are:
\n\"\"
\na)
\nThe intervals that the function is decreasing are: 3, 7, 11
\nb)
\nThe decreasing intervals are all decreasing in nature and have a negative slope<\/p>\n

c. How are the decreasing intervals different?
\nAnswer:
\nThe sizes of the decreasing intervals are different
\nSo,
\nThe values of the negative slopes for decreasing intervals will also be different<\/p>\n

Question 12.
\nCritique Reasoning<\/strong> The graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals. Do you agree? Explain.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the speed of a person riding his stationary exercise bicycle over time. a. A student claims that the function is constant in two intervals.
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nSo,
\nFrom the given graph,
\nWe can observe that
\nThe number of intervals that are constant in nature is: 4
\nHence, from the above,
\nWe can conclude that the claim of the student is not correct<\/p>\n

b. What error might the student have made?
\nAnswer:
\nThe student considered only the highest constant intervals but not the lowest intervals
\nThe student also did not consider the last constant interval due to the misconception that it is decreasing in nature but it is constant after that decreasing in nature
\nSo,
\nThe above are the errors the student has made<\/p>\n

Question 13.
\nLook for Relationships<\/strong> The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nFirst, the speed of a roller coaster increases with time, and then it fluctuates between increasing and decreasing as time increases further, and at last, the speed of the roller coaster decreases<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 14.
\nWhich statements about the graph are true? Select all that apply.
\n\"Envision The graph is decreasing in intervals (1) and (4).
\n\"Envision The graph shows a constant function in interval (2).
\n\"Envision The graph is increasing in intervals (2) and (4).
\n\"Envision The graph has a constant rate of change.
\n\"Envision The graph shows a constant function in interval (3).
\n\"Envision
\nAnswer:
\nLet the given options be named as A, B, C, D, and E respectively
\nNow,
\nThe given graph is:
\n\"Envision
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
\nHence, from the above,
\nWe can conclude that the options that match with the given situation are: A and E<\/p>\n

Lesson 3.6 Sketch Functions from Verbal Descriptions<\/h3>\n

ACTIVITY<\/strong><\/p>\n

Explain It!<\/strong><\/p>\n

The Environmental Club is learning about oil consumption and energy conservation around the world. Jack says oil consumption in the United States has dropped a lot. Ashley says oil consumption in China is the biggest problem facing the world environment.<\/p>\n

\"Envision<\/p>\n

A. Do you agree or disagree with Jack’s statement? Construct an argument based on the graph to support your position.
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nThe statement of Jack is:
\nOil consumption in the United States has dropped a lot
\nNow,
\nFrom the given graph,
\nWe can observe that the oil consumption of the United States (The first line in the graph) is constant from 2000 to 2003 and the consumption is constant from 2004 to 2008 and the consumption decreased abruptly from 2008 to 2011
\nHence, from the above,
\nWe can conclude that we can agree with Jack’s statement<\/p>\n

B. Do you agree or disagree with Ashley’s statement? Construct an argument based on the graph to support your position.
\nAnswer:
\nThe given graph is:
\n\"Envision
\nNow,
\nThe statement of Ashley is:
\nOil consumption in China is the biggest problem facing the world environment.
\nNow,
\nFrom the given graph,
\nWe can observe that the oil consumption of China (The third line in the graph) is constant from 2000 to 2004 and the consumption is increased abruptly from 2004 to 2008 and the consumption increased minimally from 2008 to 2011
\nHence, from the above,
\nWe can conclude that we can agree with Ashley’s statement since China’s consumption of oil will make the other countries suffer<\/p>\n

Focus on math practices<\/strong>
\nLook for Relationships What trend do you see in oil consumption in the United States and in Europe?
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nThe trend in the oil consumption of the United States and Europe (The first line and the second line in the graph) is:
\na) The consumption increases gradually for both the United States and Europe from 2000 to 2003
\nb) From 2004 to 2008, the oil consumption of the United States becomes constant but the consumption of Europe increases gradually
\nc) From 2009 to 2011, the oil consumption of the United States and Europe decreases gradually<\/p>\n

? Essential Question<\/strong>
\nHow does the sketch of a graph of a function help describe its behavior?
\n\"Envision<\/p>\n

Answer:
\nFrom the graph,
\nWe can observe that,
\nAt t = 0,
\nThe level of oxygen is full
\nAt t = 22 minutes,,
\nThe level of oxygen decreases
\nHence, from the above,
\nWe can conclude that
\nThe behavior of the time and the oxygen level is:
\nTime \u221d \\(\\frac{1}{Oxygen level}\\)<\/p>\n

Try It!<\/strong><\/p>\n

The weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi. Complete the statements, and then sketch the qualitative graph of this function.
\n\"Envision
\nThe input, or x-variable, is \"Envision
\nThe output, or y-variable, is \"Envision
\nAnswer:
\nIt is given that
\nThe weight of the water exerts pressure on a diver. At a depth of 10 feet, the water pressure is 19.1 pounds per square inch (psi) and at a depth of 14 feet, the water pressure is 20.9 psi.
\nSo,
\nFrom the above statements,
\nWe can observe that
\nAs the depth of the water increases, the water pressure also increases
\nSo,
\nThe input or x-variable for the given situation is: Depth
\nThe output or y-variable for the given situation is: Water pressure
\nNow,
\nTo draw the graph,
\nThe required points are: (10, 19.1), and (14, 20.9)
\nWe know that,
\nthe equation of the straight line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{20.9 – 19.1}{14 – 10}\\)
\nm = \\(\\frac{1.8}{4}\\)
\nm = \\(\\frac{9}{20}\\)
\nSo,
\ny = \\(\\frac{9}{20}\\)x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n19.1 = \\(\\frac{9}{20}\\) (10) + c
\nc = 19.1 – 4.5
\nc = 14.6
\nSo,
\ny = \\(\\frac{9}{20}\\)x + 14.6
\nHence,
\nThe representation of the above equation in the coordinate plane is:
\n\"\"<\/p>\n

Convince Me!<\/strong>
\nGeneralize How are the sketches of the two functions similar? How are they different?
\nAnswer:
\nFrom Example 1 and the above item’s graph,
\nWe can observe that both graphs have the equation in the slope-intercept form
\nBut,
\nThe graph in Example 1 is decreasing in nature
\nThe graph in the above item is increasing in nature<\/p>\n

Try It!<\/strong>
\nHaru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes. Sketch a graph of the relationship of Haru’s distance from home over time.
\n\"Envision
\nAnswer:
\nIt is given that
\nHaru rides his bike from his home for 30 minutes at a fast pace. He stops to rest for 20 minutes and then continues in the same direction at a slower pace for 30 more minutes.
\nHence,
\nThe graph of the relationship of Haru’s distance from the house over time is:
\n\"\"<\/p>\n

KEY CONCEPT<\/strong><\/p>\n

You can sketch a graph of a function to describe its behavior. When sketching a function, follow these steps:
\n1. Identify the two variables (input, output) that have a relationship.
\n2. Analyze the situation. Look for keywords that indicate that the function is increasing, decreasing, or constant.
\n3. Sketch the graph.
\n\"Envision<\/p>\n

Do You Understand?<\/strong><\/p>\n

Question 1.
\n? Essential Question<\/strong> How does the sketch of a graph of a function help describe its behavior?<\/p>\n

\"Envision<\/p>\n

Answer:
\nFrom the graph,
\nWe can observe that,
\nAt t = 0,
\nThe level of oxygen is full
\nAt t = 22 minutes,,
\nThe level of oxygen decreases
\nHence, from the above,
\nWe can conclude that
\nThe behavior of the time and the oxygen level is:
\nTime \u221d \\(\\frac{1}{Oxygen level}\\)<\/p>\n

Question 2.
\nMake Sense and Persevere<\/strong> How do you know which variable goes with which axis when you graph?
\nAnswer:
\nThe independent\u00a0variable\u00a0belongs on the x-axis\u00a0(horizontal line) of the\u00a0graph\u00a0and the dependent\u00a0variable\u00a0belongs on the y-axis\u00a0(vertical line).<\/p>\n

Question 3.
\nReasoning<\/strong> How can you determine the shape of a graph?
\nAnswer:
\nThe four ways to describe shape are whether it is symmetric, how many peaks it has if it is skewed to the left or right, and whether it is uniform. A\u00a0graph\u00a0with a single peak is called unimodal. A single peak over the center is called bell-shaped. And, a\u00a0graph\u00a0with two peaks is called bimodal.<\/p>\n

Do You Know How?<\/strong><\/p>\n

Question 4.
\nA class plants a tree. Sketch the graph of the height of the tree over time.
\n\"Envision
\na. Identify the two variables.
\nAnswer:
\nIt is given that a class plants a tree
\nNow,
\nFrom the given figure,
\nWe can observe that
\nAs time increases, the height of the tree increases
\nHence, from the above,
\nWe can conclude that
\nInput variable (or) x-coordinate: Time
\nOutput variable (or) y-coordinate: Height of the tree<\/p>\n

b. How can you describe the relationship between the two variables?
\nAnswer:
\nFrom part (a),
\nWe can observe that
\nAs time increases, the height of the tree increases
\nHence, from the above,
\nWe can conclude that the relationship between the variables is:
\nTime \u221d Height of the tree<\/p>\n

c. Sketch the graph.
\n\"Envision
\nAnswer:
\nTo draw the graph,
\nThe required points are: (0, 3), and (3, 7)
\nWe know that,
\nthe equation of the straight line in the slope-intercept form is:
\ny = mx + c
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{7 – 3}{3 – 0}\\)
\nm = \\(\\frac{4}{3}\\)
\nSo,
\ny = \\(\\frac{4}{3}\\)x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n3 = \\(\\frac{4}{3}\\) (0) + c
\nc = 3 – 0
\nc = 3
\nSo,
\ny = \\(\\frac{4}{3}\\)x + 3
\nHence,
\nThe representation of the above equation in the coordinate plane is:
\n\"\"<\/p>\n

Question 5.
\nAn airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands. Sketch the graph of the height of the plane over time.
\n\"Envision
\nAnswer:
\nIt is given that
\nAn airplane takes 15 minutes to reach its cruising altitude. The plane cruises at that altitude for 90 minutes, and then descend for 20 minutes before it lands.
\nHence,
\nThe graph of the height of the plane over time is (Height of the plane is any value) is:
\n\"\"<\/p>\n

Practice & Problem Solving<\/strong><\/p>\n

Question 6.
\nWhat relationship between money (in dollars) and time (in months) does this graph show? Write a description of the given graph.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nThe amount of money is in an increasing trend for the first half of the month
\nThe amount of money is in a decreasing trend for the second half of the month<\/p>\n

Question 7.
\nWhen a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly. Sketch the graph that represents this function over the six weeks.
\n\"Envision
\nAnswer:
\nIt is given that
\nWhen a new laptop became available in a store, the number sold in the first week was high. Sales decreased over the next two weeks and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly
\nHence,
\nThe graph which represents the given function over the six weeks is:
\n\"\"<\/p>\n

Question 8.
\nAaron’s mother drives to the gas station and fills up her tank. Then she drives to the market. Sketch the graph that shows the relationship between the amount of fuel in the gas tank of her car and time.
\n\"Envision
\nAnswer:
\nIt is given that
\nAaron’s mother drives to the gas station and fills up her tank. Then she drives to the market.
\nSo,
\nAt first, the amount of fuel in the gas tank will be of some initial value and it is in an increasing trend after filling up the tank
\nAfter that, when she drives to the market, the amount of fuel will be in a decreasing trend over time
\nHence,
\nThe graph that shows the relationship between the amount of fuel in the gas tank of her car and time is:
\n\"\"<\/p>\n

Question 9.
\nMelody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes. Sketch a graph that represents this description.
\n\"Envision
\nAnswer:
\nIt is given that
\nMelody starts at her house and rides her bike for 10 minutes to a friend’s house. She stays at her friend’s house for 60 minutes
\nHence,
\nThe graph that represents the above description is:
\n\"\"<\/p>\n

Question 10.
\nWhich description best represents the graph shown?
\nA. People are waiting for a train. A train comes and some people get on. The other people wait for the next train. As time goes by, people gradually leave the station.
\nB. One train arrives and some people get off the train and wait in the station.
\nC. People are waiting for a train. Everyone gets on the first train that comes.
\nD. People are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.
\n\"Envision
\nAnswer:
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that
\nAt first, there are some people and after some time, the number of people decreased
\nAfter some time, there are some people other than the people that decreased and after some further time, the total number of people becomes zero
\nHence, from the above,
\nWe can conclude that option D matches the above description<\/p>\n

Question 11.
\nA baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours. Sketch the graph of the relationship between the number of cakes made and time.
\n\"Envision
\nAnswer:
\nIt is given that
\nA baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours.
\nSo,
\nThe initial value for the given graph is: 10
\nHence,
\nThe graph of the relationship between the number of cakes made and time is:
\n\"\"<\/p>\n

Question 12.
\nModel with Math<\/strong> An air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple seconds until a fan grabs it. Sketch the graph that represents this situation.
\n\"Envision
\nAnswer:
\nIt is given that
\nAn air cannon launches a T-shirt upward toward basketball fans. It reaches a maximum height and then descends for a couple of seconds until a fan grabs it
\nWe know that,
\nThe projectile (Graph) of an air cannon is always like a “Parabola”
\nHence,
\nThe graph that represents the given situation is:
\n\"\"<\/p>\n

Question 13.
\nHigher-Order Thinking<\/strong> Write a verbal description of how these two variables are related. The description must suggest at least two intervals. Sketch the graph that represents the verbal description.
\n\"Envision
\nAnswer:
\nThe verbal description of the relationship between the total people and time in hours is:
\nPeople are waiting for a train. A train comes and some people get on the train. The other people wait for the next train. Another train arrives and all of the remaining people get on.
\nHence,
\nThe graph that describes the above verbal description is:
\n\"\"<\/p>\n

Assessment Practice<\/strong><\/p>\n

Question 14.
\nA baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings. Let x represent the innings of the game, and y represent the total number of runs.
\nPART A<\/strong>
\nSketch the graph of this situation below.
\n\"Envision
\nAnswer:
\nIt is given that
\nA baseball team scores the same number of runs in each of the first 4 innings. After that, the team did not score a run for the rest of the game, which lasts 9 innings.
\nHence,
\nThe graph for the above situation is:
\n\"\"
\nPART B<\/strong>
\nHow would the graph change if the innings in which the team scores runs changes?
\nAnswer:
\nIf the innings in which the team scores runs change, then
\nThe graph will be either in an increasing trend or in a decreasing trend<\/p>\n

Topic 3 REVIEW<\/h3>\n

? Topic Essential Question<\/strong><\/p>\n

How can you use functions to model linear relationships?
\nAnswer:
\nAnother approach to representing\u00a0linear functions\u00a0is by\u00a0using function\u00a0notation. One example of\u00a0function\u00a0notation is an\u00a0equation\u00a0written in the form known as the slope-intercept form of a line, where x is the input value, m is the rate of change, and b is the initial value of the dependent variable.<\/p>\n

Vocabulary Review<\/strong><\/p>\n

Match each vocabulary term with its definition.
\n\"Envision
\nAnswer:
\nEach vocabulary term with its definition is:
\n\"\"<\/p>\n

Use Vocabulary in Writing<\/strong><\/p>\n

Explain how to write a linear function in the form y = mx + b by using the two points given below. Use vocabulary words in your explanation. (0, -2), (2, 6)
\nAnswer:
\nThe given points are:
\n(0, -2) and (2, 6)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{6 + 2}{2 – 0}\\)
\nm = \\(\\frac{8}{2}\\)
\nm = 4
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = 4x + c
\nNow,
\nTo find the value of c,
\nSubstitute any one of the points in the above equation
\nSo,
\n-2 = 0 + c
\nc = -2 – 0
\nc = -2
\nHence, from the above,
\nWe can conclude that the representation of the linear equation for the given points is:
\ny = 4x – 2<\/p>\n

Concepts and Skills Review<\/strong><\/p>\n

LESSON 3.1 Understand Relations and Functions<\/strong><\/p>\n

Quick Review<\/strong>
\nA relation is a set of ordered pairs. A relation is a function if each input, or x-value, has exactly one unique output, or y-value.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nIs the relation shown in the table a function? Explain.
\n\"Envision
\nAnswer:
\nThe given relation is:
\n\"Envision
\nWe know that,
\nA relation is said to be a function if each input has only 1 output
\nSo,
\nFrom the given relation,
\nWe can observe that each input has only 1 output
\nHence, from the above,
\nWe can conclude that the given relation is a function<\/p>\n

Question 2.
\nDoes the relation {(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)} represent a function? Use the arrow diagram. Then explain your answer.
\n\"Envision
\nAnswer:
\nThe given ordered pairs are:
\n(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)
\nWe know that,
\nThe ordered pairs are in the form of (input (x), output (y))
\nIn the arrow diagram,
\nThe left side represents the values of the input
\nThe right side represents the values of the output
\nSo,
\nThe representation of the ordered pairs in the arrow diagram is:
\n\"\"
\nWe know that,
\nA relation is said to be a function only when each input corresponds with only 1 output
\nNow,
\nFrom the arrow diagram,
\nWe can observe that
\nThe same input corresponds with 2 outputs
\nHence, from the above,
\nWe can conclude that the given relation is not a function<\/p>\n

LESSON 3.2 Connect Representations of Functions<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can represent a function in a table, in a graph, or as an equation. The graph of a linear function is a straight line.<\/p>\n

Practice<\/strong>
\nMark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
\n1. Write an equation that relates the amount left on the card, y, over time, x.
\nAnswer:
\nIt is given that
\nMark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
\nNow,
\nLet y be the amount left on the card
\nLet x be the number of weeks
\nSo,
\nThe amount left on the card = The total amount of the card – The amount of money he used to buy a new app for x weeks
\ny = $100 – $4.99x
\nHence, from the above,
\nWe can conclude that
\nThe equation that relates the amount left on the card, y, over time, x is:
\ny = $100 – $4.99x<\/p>\n

2. Make a graph of the function.
\n\"Envision
\nAnswer:
\nFrom part (a),
\nThe equation that relates the amount left on the card, y, over time, x is:
\ny = $100 – $4.99x
\nHence,
\nThe representation of the above equation in the coordinate plane is:
\n\"\"<\/p>\n

LESSON 3.3 Compare Linear and Nonlinear Functions<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can compare functions in different representations by looking at the properties of functions: the constant rate of change and the initial value.<\/p>\n

Practice<\/strong>
\nTwo linear functions are shown.
\n\"Envision
\nAnswer:
\nThe given linear functions are:
\n\"Envision
\nFunction A:
\nCompare the given equation with
\ny = mx + c
\nWhere,
\nm is the rate of change (or) slope
\nc is the initial value (or) y-intercept
\nSo,
\nFrom the above equation,
\nThe rate of change is: -3
\nThe initial value is: 2
\nFunction B:
\nFrom the given table,
\nWe can observe that
\nThe rate of change for all the values of x is: 1
\nThe rate of change for all the values of y is: 2
\nSo,
\nThe rate of change for the table = \\(\\frac{y}{x}\\)
\n= \\(\\frac{2}{1}\\)
\n= 2
\nWe know that,
\nThe initial value is the value of y when x = 0
\nSo,
\nFor the given table,
\nThe initial value is: 0
\nThe rate of change is: 2<\/p>\n

Question 1.
\nWhich function has the greater initial value? Explain.
\nAnswer:
\nFunction A has the greater initial value<\/p>\n

Question 2.
\nWhich function has the greater rate of change?
\nAnswer:
\nFunction B has the greater rate of change<\/p>\n

LESSON 3.4 Construct Functions to Model Linear Relationships<\/strong><\/p>\n

Quick Review<\/strong>
\nA function in the form y = mx + b represents a linear relationship between two quantities, x and y, where m represents the constant rate of change and b represents the initial value.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nWhat is the equation of a line that passes through (0.5, 4.25) and (2, 18.5) and has a y-intercept of -0.5?
\nAnswer:
\nIt is given that
\nA line passing through (0.5, 4.25), and (2, 18.5) and has a y-intercept of\u00a0 -0.5
\nWe know that,
\nThe y-intercept is the value of y when x = 0
\nNow,
\nThe given points are:
\n(0.5, 4.25) and (2, 18.5)
\nNow,
\nWe know that,
\nSlope (m) = y2<\/sub> – y1<\/sub> \/ x2<\/sub> – x1
\n<\/sub>So,
\nCompare the given points with (x1<\/sub>, y1<\/sub>), (x2<\/sub>, y2<\/sub>)
\nSo,
\nm = \\(\\frac{18.5 – 4.25}{2 – 0.5}\\)
\nm = \\(\\frac{14.25}{1.5}\\)
\nm = 9.5
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = 9.5x – 0.5
\nHence, from the above,
\nWe can conclude that the representation of the linear equation for the given points is:
\ny = 9.5x – 0.5<\/p>\n

Question 2.
\nThe graph shows the relationship of the number of gallons being drained from an aquarium over time. What function models the relationship?
\n\"Envision
\nAnswer:
\nIt is given that
\nThe graph shows the relationship of the number of gallons being drained from an aquarium over time.
\nNow,
\nThe given graph is:
\n\"Envision
\nFrom the given graph,
\nWe can observe that there is an initial value or y-intercept
\nSo,
\nThe y-intercept from the given graph is: 90
\nNow,
\nTo find the slope from the given graph,
\nThe given points are: (0, 90), and (9, 0)
\nSo,
\nSlope (m) = \\(\\frac{0 – 90}{9 – 0}\\)
\nm = \\(\\frac{-90}{9}\\)
\nm = -10
\nWe know that,
\nThe representation of the equation in the slope-intercept form is:
\ny = mx + c
\nSo,
\ny = -10x + 90
\nHence, from the above,
\nWe can conclude that the equation that represents the given relationship is:
\ny = -10x + 90<\/p>\n

LESSON 3-5 Intervals of Increase and Decrease<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can describe the relationship between two quantities by looking at the behavior of the line at different intervals on a qualitative graph. The function is increasing if both x- and y-values increase. The function is decreasing if the y-values decrease as the x-values increase.<\/p>\n

Practice<\/strong><\/p>\n

The graph shows the altitude of an airplane over time.
\n\"Envision
\nAnswer:
\nWe know that,
\nThe graph is said to be increasing in nature when the line moves from left to right or the slope is positive
\nThe graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
\nThe graph is said to be constant in nature when the line is parallel to any one of the coordinate axes<\/p>\n

Question 1.
\nIn which intervals is the graph of the function constant? Explain.
\nAnswer:
\nThe intervals in which the graph of the function is constant are: 2, 4, 6<\/p>\n

Question 2.
\nIn which intervals is the graph of the function decreasing? Explain.
\nAnswer:
\nThe intervals in which the graph of the function is decreasing in nature are: 5, 7<\/p>\n

LESSON 3.6 Sketch Functions from Verbal Descriptions<\/strong><\/p>\n

Quick Review<\/strong>
\nYou can sketch a graph of a function to describe its behavior. When sketching a function, identify the variables (input, output) that have a relationship, analyze the situation, and then sketch the graph.<\/p>\n

Practice<\/strong><\/p>\n

Question 1.
\nJack’s mother brings him a bowl of carrots as a snack. At first he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat any more. Sketch a graph that shows the number of carrots in the bowl over time.
\n\"Envision
\nAnswer:
\nIt is given that
\nJack’s mother brings him a bowl of carrots as a snack. At first, he does not eat any; then he eats one at a time until half of the carrots are gone. Then he does not eat anymore
\nSo,
\nAt first,
\nThe number of carrots present in a bowl is: An arbitrary number
\nThe number of carrots Jack consumed is: 0
\nHence,
\nThe graph that shows the number of carrots in the bowl over time is:
\n\"\"<\/p>\n

Topic 3 Fluency Practice<\/h3>\n

What’s the Message?<\/strong><\/p>\n

In each row, determine which equation has the greater solution. Circle the letter next to the equation with the greater solution in each row.<\/p>\n

\"Envision
\nAnswer:
\nThe row of equations that has great solutions are:
\n\"\"<\/p>\n

What does the zero say to the eight?
\n\"Envision
\nAnswer:
\nThe statement said by zero to eight is:
\nYou are greater than me<\/p>\n","protected":false},"excerpt":{"rendered":"

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