Practice with the help of enVision Math Common Core Grade 8 Answer Key Topic 3 Use Functions to Model Relationships regularly and improve your accuracy in solving questions.
enVision Math Common Core 8th Grade Answers Key Topic 3 Use Functions To Model Relationships
Topic 3 GET READY!
Review What You Know!
Choose the best term from the box to complete each definition.
The ____ is the ratio of the vertical change to the horizontal change of a line.
We know that,
The "Slope" is the ratio of the vertical change to the horizontal change of a line
Hence, from the above,
We can conclude that the best term from the box to complete the given definition is "Slope"
A relationship that can be modeled by the equation y = mx is a ___
A relationship that can be modeled by the equation y = mx is a "Proportional relationship"
We can conclude that the best term from the box to complete the given definition is "Proportional relationship"
y-value at which a line of a graph crosses the y-axis is called the ___
y-value at which a line of a graph crosses the y-axis is called the "y-intercept"
We can conclude that the best term from the box to complete the given definition is "y-intercept"
An equation written in the form y = mx + b is called the ___
An equation written in the form y = mx + b is called the "Linear equation" or the "Slope-intercept form"
We can conclude that the best term from the box to complete the given definition is "Linear equation"(or) the "Slope-intercept form"
Slope and y-Intercept
Find the slope and y-intercept of a line that passes through these points.
(2, 2) and (3, 0)
The given points are:
(2, 2), and (3, 0)
Compare the given points with (x1, y1), (x2,y2)
Slope(m) = y2 – y1 / x2 – x1
m = \(\frac{0 – 2}{3 – 2}\)
= \(\frac{-2}{1}\)
= -2
The linear equation in the slope-intercept form is:
y = mx + c
m is the slope
c is the y-intercept
We can obtain the y-intercept by putting the value of x equal to 0
y = -2x + c
Substitute (3, 0) or (2, 2) in the above equation
0 = -6 + c
c = 6
We can conclude that
The slope of a line that passes through the given points is: -2
The y-intercept of a line is: 6
(1, 5) and (4, 10)
(1, 5), and (4, 10)
m = \(\frac{10 – 5}{4 – 1}\)
= \(\frac{5}{3}\)
y = \(\frac{5}{3}\)x + c
Substitute (4, 10) or (1, 5) in the above equation
5 = \(\frac{5}{3}\) + c
c = \(\frac{10}{3}\)
The slope of a line that passes through the given points is: \(\frac{5}{3}\)
The y-intercept of a line is: \(\frac{10}{3}\)
(8, 2) and (-8,6)
(8, 2), and (-8, 6)
m = \(\frac{6 – 2}{-8 – 8}\)
= \(\frac{4}{-16}\)
Substitute (-8, 6) or (8, 2) in the above equation
2 = –\(\frac{1}{4}\) (8) + c
Compare Proportional Relationships
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.
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
The unit cost rate of a bottle in store A = \(\frac{The cost of 18 bottles}{18}\)
= \(\frac{$10}{18}\)
= $0.55
The unit cost rate of a bottle in store B = \(\frac{The cost of 12 bottles}{12}\)
= \(\frac{$6}{12}\)
= $0.5
The representation of the unit cost rate of a bottle in both stores is:
From the above graph,
We can observe that
The unit cost rate of a bottle in store B < The unit cost rate of a bottle in store A
We can conclude that store B sells the drinks for less cost
Write the equation for the graph of the line shown.
The given graph is:
The y-intercept is the value of the point that passes through the y-axis
From the graph,
The point that passes through the y-axis is: (0, -6)
The y-intercept is: -6
To find the slope,
The points are: (2, 2), and (0, -6)
m = \(\frac{-6 – 2}{0 – 2}\)
= \(\frac{-8}{-2}\)
We can conclude that the equation of the line for the given graph is:
Write key words or phrases associated with each representation. Then write function or not a function on the given lines.
Topic 3 PICK A PROJECT
What machine could be invented to make your life better?
PROJECT: BUILD A RUBE GOLDBERG MACHINE
What games can you play indoors?
PROJECT: MAKE A MATH
What are the steps for fixing a leaky pipe?
PROJECT: PLAN A MAINTENANCE ROUTE
PROJECT 3D
If you were to make a video game, what kind of game would it be?
PROJECT: DESIGN A VIDEO GAME ELEMENT
Lesson 3.1 Understand Relations and Functions
Solve & Discuss It!
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.
Whose plan would you recommend? Explain.
$1 1 ticket
$5 6 tickets
$20 25 tickets
It is given that
The 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.
It is also given that
$1 – 1 ticket
$5 – 6 tickets
$20 – 25 tickets
According to Jesse's goal,
Each member should sell 50 raffle tickets to make the total amount of $500
The possible combinations may be:
50 $1 tickets will be sold by each member
Any other combination is not possible
According to Alexis's plan,
Each member should raise $50 to make the total amount of $500
It is possible and very easy because $50 by each member can be raised in many ways
We can conclude that Alexis's plan would be recommended
Focus on math practices
Reasoning How are the two plans different? How are they similar?
? Essential Question
when is a relation a function?
A "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.
Joe 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).
(4, 24), (5, 35), (8, 24), (2, 20), (9, 27)
Complete the arrow diagram. Is the area of a brochure a function of the side length? Explain.
The ordered pairs can be represented in the form of (x, y)
x is the input
y is the output
From the given ordered pairs,
We can observe that for different values of the input, there are different values of output
If there are the same outputs for the different inputs, then also a relationship is considered a function
The complete arrow diagram for the given ordered pairs are:
We can conclude that the area of the brochure is a function of the side length
Convince Me!
There are two outputs of 24. Does this help you determine whether the relation is a function? Explain.
Hence, from the above
We can conclude that even for the two outputs of 24, the given relationship is considered a function
Frank reverses the ordered pairs to show the heights and ages of the same six students. Is age a function of height? Explain.
Frank reverses the ordered pairs to show the heights and ages of the same six students.
The given table is:
From the given table,
For different values of age, there are different values of height
Age —-> Input
Height —> Output
We can conclude that age is a function of height
Heather 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.
Heather claims that she can tell exactly how long a family was at the museum by how much the family pays for parking.
The table for the given situation is: (Example 3)
From the table,
There are different costs for the different times in hours
We can say that cost is a function of time
We can conclude that Heather's claim is correct
KEY CONCEPT
A 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.
This relation is a function.
This relation is not a function.
A 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
How can you use different representations of a relation to determine whether the relation is a function?
Relations 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 an ordered pair: Displaying a relation as a table
Is a relation always a function? Is a function always a relation? Explain.
All 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.
Is the relation shown below a function? Explain.
The given relation is:
From the given relation,
We can observe that there is the same input for the different outputs,
A relation can be considered as a function when the different inputs have different outputs
We can conclude that the given relation is not a function
We can observe that there are different outputs for different inputs
We can conclude that the given relation is a function
(4,16), (5, 25), (3,9), (6, 36), (2, 4), (1, 1)
Practice & Problem Solving
The 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.
a. Make an arrow diagram that represents
The arrow diagram for the given relation is:
b. is the relation a function? Explain.
Does the relation shown below represent a function? Explain.
(-2, 2), (-7, 1), (-3, 9), (3, 4), (-9,5), (-6, 8)
Is the relation shown in the table a function? Explain.
We can observe that there are different outputs for the same inputs
Construct Arguments
During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in °C).
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100) Is the relation a function? Explain.
(0, 15), (5, 20), (10,50) (15, 80). (20, 100), (25, 100)
An ordered pair can be represented as (x, y)
x is the time
y is the temperature in °C
We can observe that there are different outputs for the different inputs
Taylor 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.
Taylor has tracked the number of students in his grade since third grade. He records his data in the table
We can observe that there are different outputs (People) for the different inputs (Grade)
We can conclude that the given table is a function
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week. Is this relation a function? Explain.
James raises chickens. He tracks the number of eggs his chickens lay at the end of each week
We can observe that there are different outputs (Eggs) for the different inputs (Weeks)
Relations P and Q are shown below.
a. Make an arrow diagram to represent Relation P.
Hence,
The arrow diagram to represent the relation P is:
b. Make an arrow diagram to represent Relation Q.
The arrow diagram to represent the relation Q is:
c. Which relation is a function? Explain.
From relation P,
From relation Q,
We can conclude that relation P is a function
Higher-Order Thinking
On 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?
On 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.
From the given arrow diagram and ordered pairs,
In the arrow diagram, inputs and outputs are reversely represented
We can conclude that the error Bobby most likely made is the reversal of inputs and outputs
Write the set of ordered pairs that is represented by the arrow diagram at the right. Is the relation a function? Explain.
The given arrow diagram is:
In the arrow diagram,
The left side represented the inputs and the right side represented the outputs
The representation of the arrow diagram in the form of the ordered pairs (Input, Output) are:
(49, 13), (61, 36), (10, 27), (76, 52), (23, 52)
From the above relation,
Which of these relations are functions? Select all that apply.
From the given relations,
Relation 2, Relation 3 are the functions
We can conclude that Relation 2 and Relation 3 are the functions
Lesson 3.2 Connect Representations of Functions
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?
Construct Arguments How does finding an average distance the whales travel in miles help with finding a solution to this problem?
What are different representations of a function?
Relationships and functions can be represented as graphs, tables, equations, or verbal descriptions. Each representation gives 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.
As 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.
The amount of water remaining in the pool is gallons.
The amount of water pumped each hour is gallons.
The equation is
As the pump is pumping water, the amount of water in the pool decreases at a constant rate
Let the initial amount of water present in the pool is: 9,000 gallons
The rate of the amount of water that pumped each hour = \(\frac{The initial amount of water present in the pool}{The time that is present where the initial amount of water present}\)
= \(\frac{9,000}{12}\)
= 750 gallons per hour
The amount of water remaining in the pool = The initial amount of water present in the pool – The amount of water that pumped each hour
= 9,000 – 1,500
= 7,500 gallons
Let the number of hours be x
The linear equation is in the form of
The total amount of water present in the pool = The rate at which the water pumps out + The amount of water that pumped each hour
9,000 = 750x + 7,500
The amount of water remaining in the pool is 7,500 gallons.
The amount of water pumped each hour is 1,500 gallons.
The equation is:
How is the rate of change of this function different from that in Example 1? Explain.
The 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
Draw a graph that represents a linear function?
The representation of the linear equation is:
y = mx —–> Slope form
y = mx + c —-> Slope-intercept form
Let the linear equation in the slope-intercept form be:
y = x + 3
The graph of the above linear equation in the coordinate plane is:
You can represent a function in different ways: in a table, in a graph, or as an equation.
A day at the amusement park costs $10 for an entrance fee and $2.50 for each ride ticket.
?Essential Question What are different representations of a function?
Use Appropriate Tools How can you use a graph to determine that a relationship is NOT a function?
Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.
Construct Arguments Must the ordered pairs of a function be connected by a straight line or a curve on a graph? Explain.
The points can be connected by a straight line. Thus, the ordered pairs represent a linear function.
Each 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.
Each week, Darlene tracks the number of party hats her company has in stock. The table shows the weekly stock.
A relation is defined as a function only when there are different outputs for different inputs
We can observe that the outputs (party hats) are different for different inputs (Weeks)
The given function can be called "Linear function" if the rate of change is constant
The given function can be called a "Non-linear function" if the rate of change is not constant
The rate of change = Initial value – Next value
We can conclude that the given relationship is a linear function
How can Darlene use the graph above to know when to order more party hats?
Darlene know when to order more hats when there is no stack when observing the track sheet
Leveled Practice In 6-7, explain whether each graph represents a function.
We can observe that each input has a different output and the rate of change is constant
We can conclude that the given graph represents a function
From the given graph,
We can observe that each input has a different output but the rate of change is not constant
We can conclude that the given graph represents a function but a non-linear function
Hannah approximates the areas of circles using the equation A = 3r2 and records areas of circles with different radius lengths in a table.
a. Graph the ordered pairs from the table.
The representation of the ordered pairs (in, in²) are:
(1, 3), (2, 12), (3, 27), (4, 48), (5, 75)
The representation of the ordered pairs in the coordinate plane is:
From part (a),
The 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.
The relationship between the number of hexagons, x, and the perimeter of the figure they form, y, is shown in the graph
We can observe that for different values of perimeters, the number of hexagons is also different
We can conclude that the perimeter of a figure is a function of the number of hexagons
Construct Arguments Do the ordered pairs plotted in the graph below represent a function? Explain.
We can observe that for the different values of x, there are different values of y
We can conclude that the given ordered pairs in the graph represents a function
A train leaves the station at time t = 0. Traveling at a constant speed, the train travels 360 kilometers in 3 hours.
a. Write a function that relates the distance traveled, d, to the time, t.
Speed = \(\frac{Distance}{Time}\)
Speed is constant
Distance = m (Time)
d = mt
m is the rate of change or proportionality constant
m = \(\frac{Distance}{Time}\)
= \(\frac{360}{3}\)
= 120 km / hour
We can conclude that the function that relates to the distance d, and time t is:
d = 120t
b. Graph the function and tell whether it is a linear function or a nonlinear function.
The function is a function.
The function that relates the distance d, and time t is:
Compare the above equation with y = mx
The graph of the above function is:
We can conclude that the given function is a linear function from the above graph
Higher-Order Thinking Tell whether each graph is a function and justify your answer. Which graph is not a good representation of a real-world situation? Explain.
The given graphs are:
From graph A,
We can observe that there are different outputs for the same input
Graph A does not represent the function
From graph B,
Graph B does represent a function
We can say that graph A does not represent the real-world situation
You have an ant farm with 22 ants. The population of ants on your farm doubles every 3 months.
Complete the table.
The completed table for the given situation is:
Is the relation a function? If so, is it a linear function or a nonlinear function? Explain.
We can observe that the ant population is different for the different number of months
Rate of change = \(\frac{44}{22}\)
The rate of change is also constant for all the table
We can conclude that the given relation is a linear function
Use the function y = \(\frac{3}{2}\)x + 3 to complete the table of values.
The given function is:
y = \(\frac{3}{2}\)x + 3
The completed table for the given values of y is:
Lesson 3.3 Compare Linear and Nonlinear Functions
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?
Two streaming video subscription services offer family plans with different monthly costs, as shown in the ads
Let the number of devices be x
Let the total subscription cost be y
For Movies 4 U,
The total subscription cost = The subscription cost of the first device + The subscription fee for additional devices
y = $10 + $2x
For Family Stream,
y = \(\frac{$12}{4}\) + $1x
y = $3 + $1x
The above equations are in the form of slope-intercept form
The slope-intercept form of the equation is:
The common points in the two plans are:
A) The subscription cost of the first device plan
B) The additional fee plan
The different points in the two plans are:
A) The first plan consists of the additional fee of each device
B) The second plan consists of the additional fee for greater than 5 devices
Movies 4 U is better than Family Stream when the subscription cost of the first device will be less
How can you represent the relationship between cost and number of devices?
The representation of the relationship between the cost and the number of devices is:
Look for Relationships Describe the relationship between the cost and the number of devices for each service. What do you notice about each relationship?
The relationship between the cost and the number of devices for each service is:
In the service of Family Stream,
The subscription fee is given for up to 4 devices and the additional subscription fee is for greater than 5 devices
How can you compare two functions?
The two functions can be compared by:
A) Identify the rate of change for the first function
B) Identify the rate of change for the second function
C) Identify the y-intercept of the first function
D) Identify the y-intercept of the second function
E) Compare the properties of each function
The 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?
The 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)
For a third robot,
The wielding rate = \(\frac{The number of wielding tasks (w)}{Time in minutes (t)}\)
= 10.8 (From the equation t = 10.8w)
When we compare the wielding rates of the three robots,
The 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)
We can conclude that the comparison of the wielding rates of the three robots is:
The first robot's wielding rate < The third robot's wielding rate < The second robot's wielding rate
How can linear equations help you compare linear functions?
While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear (a straight line) and a function (matching each x-value to only one y-value).
Compare the properties of these two linear functions.
The given functions are:
For function 1,
Find out the rate of change and the y-intercept i.e., the initial value
Rate of change = \(\frac{y}{x}\)
For y = 1 and x = 2.
Rate of change = 0.5
For y = 5.5 and x = 5,
From the above values,
We can say that the rate of change is not constant
The given function is a non-linear function and it does not have any initial value i.e., the y-intercept is 0
Compare the given equation with
m is the slope or the rate of change
c is the initial value or the y-intercept
From the given equation,
Rate of change (m): 2
Hence, from the above 2 functions,
The y-intercept of function1 > The y-intercept of function 2
You can compare functions in different representations by using the properties of functions.
Compare the constant rate of change and the initial value.
Reasoning 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?
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.
For Anne,
The rate of change is defined as the average speed
The y-intercept is defined as the beginning point
The rate of change is: 6 miles per hour
The y-intercept is: 4
For John,
Rate of change = \(\frac{y}{x}\) = \(\frac{Mile marker}{Time (hours)}\)
The total distance traveled by John (y) = Final value – Initial value
= 11.5 – 1
= 10.5 miles
The total time took by John (x) = 1.5 hours
= \(\frac{10.5}{1.5}\)
= 7 miles per hour
The y-intercept is the value of y when x = 0
By comparing the values of the rate of change,
We can conclude that John is running faster
Reasoning In Item 2, how do Anne and John's starting positions compare? Explain.
The starting positions are nothing but the initial positions of both Anne and John i.e., the y-intercepts of both Anne and John
The y-intercept of Anne is: 4
The y-intercept of John is: 1
By comparing the y-intercepts,
The starting position of Anne > The starting position of John
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.
m is the rate of change
From the given equation
For Felipe,
The rate of change is: -30
The initial value is: 290
For Samantha,
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to the value of y}\)
Whose musical instrument costs more, Felipe's or Samantha's? Explain.
If the amount owed (y) is less i.e, the rate of change is negative, then the cost of the instrument will also be less
When we compare the rate of change of Felip and Samantha,
The rate of change of Felip < The rate of change of Samantha
Hence, from the above comparison,
We can conclude that the instrument of Samantha costs more
Who will pay more each month? Explain.
We can conclude that Samantha will pay more each month
Two linear functions are shown below. Which function has the greater rate of change?
The given linear functions are:
The rate of change = \(\frac{y}{x}\)
For Function A,
The rate of change = \(\frac{Any value of y}{The value of x that corresponds to y}\)
For Function B,
We can conclude that Function A has a greater rate of change
Two linear functions are shown below. Which function has the greater initial value?
The initial value is also known as the y-intercept
The initial value (y-intercept) is: 4
We can conclude that Function A has the greater initial value
Tell whether each function is linear or nonlinear.
To find whether the given function is linear or non-linear, we have to verify whether the rate of change is constant or not
If the rate of change is constant, then the function is linear
If the rate of change is not constant, then the function is non-linear
For x = 1 and y = 2,
The rate of change = 2
The rate of change = 2.5
Function A is a non-linear function
For x = 1.5 and y = 3,
Function B is a non-linear function
For a relation to be a graph, each input has a different output but the same input will not have different outputs
From Function A,
We can observe that there are different inputs for different outputs i.e., the values of x and y are different
The rate of change is not constant since the graph is non-linear
From Function B,
The rate of change is constant for Function B
Function B is a linear function
Determine whether each function is linear or nonlinear from its graph.
If the graph is a straight line, then the function is a linear function
If the graph is not a straight line, then the function is not a linear function
The function 1 is a linear function
The function 2 is a non-linear function
Look for Relationships Justin opens a savings account with $4. He saves $2 each week. Does a linear function or a nonlinear function represent this situation? Explain.
Justin opens a savings account with $4. He saves $2 each week.
We can observe that there is a constant rate of change
For weeks,
We can obtain the number of weeks by adding 1 i.e., 0 + 1, 1 + 1, etc
The rate of change is constant i.e., 1
For money in account,
We can obtain the money by adding 2 to the initial amount of money i.e., 4 + 2, 6 + 2, etc
We can conclude that since the rate of change is constant, the given situation represents a linear function
Reasoning 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?
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.
For Player A,
The given equation is:
Compare the givene quation with
The rate of change of Player A is: 4
For Player B,
We can observe that the rate of change is constant for both the values of x and y
The rate of change for both the values of x and y is: 1
The rate of change of Player B is: 1
The rate of change of Player A > The rate of change of Player B
We can conclude that Player A earns more points
Two 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?
Two 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
For Athlete A,
The starting point is: 16
The initial point for Athlete A is: 16
For Athlete B,
The initial point or the y-intercept is the value of y when x = 0
The initial point for Athlete B is: 12
The initial points in the given situation describe the number of pushups one can do at a time without stop
The initial point for Athlete A > The initial point for Athlete B
Higher-Order Thinking 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.
The equation y = 4x – 2 and the table and graph shown at the right describe three different linear functions
m is the constant rate of change
For the given equation,
The rate of change is: 4
The given table and graph are:
For x = 2 and y = 10
Since the rate of change is constant for all the cases,
The rate of change for the given table is: 5
The given points to find the slope are: (0, 4), and (2, 0)
Slope (or) The rate of change = \(\frac{0 – 4}{2 – 0}\)
When we compare the rate of change for all the three linear functions,
The rate of change of the table > The rate of change of the equation < The rate of change of the graph
The function that has the greatest rate of change is: Table
The function that has the least rate of change is: Graph
The 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.
The 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.
The rate of change of grapes consumption is: 12 grapes per minute
The rate of change of carrots consumption = \(\frac{The difference between any 2 values of carrots remaining}{The values of the tie elapsed corresponding to the carrots remaining}\)
= \(\frac{118 – 136}{8 – 6}\)
= –\(\frac{18}{2}\)
= -9 carrots per minute
The consumption rate of grapes > The consumption rate of carrots
Hence, from the baove,
We can conclude that grapes can be eaten at a faster rate
The 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?
The initial height of Candle A is greater than the initial height of Candle B.
The height of Candle A decreases at a faster rate than the height of Candle B.
Candle B will burn out in 58 hours.
After 10 hours, the height of Candle A is 110 millimeters.
Candle A will burn out before Candle B.
Let the given options be named as A, B, C, D, and E
The 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
The rate of change of Candle A = \(\frac{201 – 177}{4}\)
= 6 millimeters per hour
For Candle B,
The time to burn the Candle B = \(\frac{290}{5}\)
= 58 hours
We can conclude that B, C, and E match with the given situation
Topic 3 MID-TOPIC CHECKPOINT
Vocabulary How can you determine whether a relation is a function? Lesson 3.1
Identify the output values. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.
Can an arrow or arrows be drawn from 10.3 so the relation in the diagram is a function? Explain your answer. Lesson 3.1
The given diagram is:
For the relation in the diagram to be a function,
The 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
Two linear functions are shown below. Which function has the greater rate of change? Justify your response. Lesson 3.3
For a linear function,
The rate of change is constant
y = \(\frac{1}{2}\)x – 1
y = 0.5x – 1
Compare the above equation with
The rate of change is: 0.5
We can conclude that the two linear functions have the same rate of change
Neil 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
Neil 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.
For the number of hours as input and the Grades as the output,
For each value of the hours, there are the same values of the Grade
The relation of hours of study time to the grade earned on a test is not considered a function
The representation of the relation in the form of the ordered pairs is:
(4, 75), (6, 75), and (6, 82)
Is the function shown linear or nonlinear? Explain your answer. Lesson 3.3
For each value of x, there is only 1 value of y
The given graph is a function
A function is called a linear if the graph is a straight line
A function is called non-linear if the graph is in any shape other than the straight line
We can conclude that the given graph is a non-linear function
How well did you do on the mid-topic checkpoint? Fill in the stars.
Topic 3 MID-TOPIC PERFORMANCE TASK
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.
The given graph is a straight line
The rate of change of the given graph = \(\frac{y}{x}\)
= \(\frac{0 – 10}{1 – 0}\)
= -10
The given graph is considered a linear function
To consider a function linear, verify whether the rate of change is constant or not
For all the values of x,
For all the values of y,
The rate of change is: 35
The rate of change of the given table = \(\frac{y}{x}\)
The given table is considered a linear function
y = 10x + 7
The rate of change for the given equation is: 10
The given equation is considered a linear function
Is each of the proposals represented by linear functions? Explain.
Yes, all the proposals are represented by linear functions
Does the class have any money in the account now? How can you tell?
We can observe that the straight line does not start from 0 but from 10
The initial value of the graph will be: 10
The initial value is considered the y-intercept
We can conclude that the class have the money in the account now i.e., $10
Which fundraising proposal raises money at the fastest rate? Explain.
Since the rate of change is the highest for Paul's proposal,
Paul's Proposal raises money at the fastest rate
If 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.
It is given that Sarah and her classmates are hoping to raise $200
To raise the money,
We have to choose the plan which has the highest rate of change
We can conclude that Sarah and her classmates choose Paul's proposal
3-Act Mathematical Modeling: Every Drop Counts
3-ACT MATH
АСТ 1
After watching the video, what is the first question that comes to mind?
After watching the video,
The first question that comes to mind is:
How much amount of water people waste brushing their teeth?
Write the Main Question you will answer.
The main question you will answer is:
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
The answer to the main question is: 4 gallons
The prediction of the answer for the main question is according to the surveys done by International Organisations
On the number line below, write a number that is too small to be the answer. Write a number that is too large.
From the above,
The maximum amount (Too large) of water used to brush teeth is: 4 gallons
The minimum amount (Too small) of water used to brush the teeth is: 2 gallons
The representation of the amounts of water used to brush teeth in this situation is:
Plot your prediction on the same number line.
We can observe that there are minimum and maximum amounts of water used to brush the teeth
Let x be the amount of water used to brush the teeth
The prediction will be: 2 < x < 4
The representation of the prediction on the number line is:
What information in this situation would be helpful to know? How would you use that information?
The information in this situation that would be helpful to know is:
How much time did it take to completely brush your teeth?
From the above information,
We can estimate the amount of water used to brush your teeth
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Represent the situation using mathematics.
Use your representation to answer the Main Question.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
Write the answer you saw in the video.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
Be Precise How do the units you chose and the method you used help you communicate your answer?
Answer;
Use Structure How much water will he save in a year?
Lesson 3.4 Construct Functions to Model Linear Relationships
Explore It!
Erick 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.
The total amount he saved = The amount he saved already + The amount he planned to save each week
The number of weeks —-> x
The total amount he saved —-> y
y = $20x + $120
Compare the above equation with y = mx + c
m s the rate of change (or) slope
The rate of change (m) is: 20
The y-intercept is: 120
A. Complete the table. Then graph the data.
The completed table for the above equation is:
The representation of the given equation in the coordinate plane is:
B. How can you tell that the relationship is a linear function from the table? How can you tell from the graph?
A relation is said to be a function when an input value matches only with an output value
A function is said to be a linear function when the graph of that function is a straight line
We can observe that for each value of the week, there is a different amount saved
The given relation is said to be a function
From the graph of the equation,
We can observe that the graph is a straight line
We can say that the function is a linear function
Generalize How can the different representations help you determine the properties of functions?
The different representations of the functions are:
A) Symbolic or Algebraic representation – The properties can be found out by comparing with the standard form
B) Numerical (Tables) representation – The properties can be found out by the rate of change and the initial values
C) Graphical representation – The properties can be found out by the values of x and y
D) Verbal representation – The properties can be found out by the keywords
How can you use a function to represent a linear relationship?
Another 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.
How 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?
Graph the function. The slope of the function shown in the graph is . The equation of the function is y =
x. If the base length is 110 inches, then the height of the ramp will be inches.
The plan shows that for every 3 inches of height, the triangle should have a base that is 15 inches long
The rate of change (m) = \(\frac{Rise}{Run}\)
m = \(\frac{3}{15}\)
m = \(\frac{1}{5}\)
The slope of the function shown in the graph is: \(\frac{1}{5}\)
y = mx
y = \(\frac{1}{5}\)x
y is the height of the ramp
x is the base length of the triangle
For the base length of 110 inches,
y = \(\frac{1}{5}\) (110)
y = 22 inches
We can conclude that the height of the ramp will be 22 inches for the base length of 110 inches
Explain why the initial value and the y-intercept are equivalent.
An equation in slope-intercept form of a line includes the slope and the initial value of the function. The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point where the line crosses the y-axis.
Jin 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.
Jin 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.
Let x be the number of weeks
Let y be the number of cups of dog food used
For x = 2, y = 8.5 (The value of 8\(\frac{1}{2}\))
For x = 5, y = 21.25 (The value of 21\(\frac{1}{4}\))
The equation in the slope-intercept form is:
8.5 = 2m + c —–> (1)
21.25 = 5m + c ——> (2)
Solve eq (1) and eq (2)
8.5 = 2m + 21.25 – 5m
8.5 – 21.25 = 2m – 5m
-12.75 = -3m
3m = 12.75
m = \(\frac{12.75}{3}\)
m = 4.25
For the value of c,
Substitute the value of m either in eq (1) or in eq (2)
8.5 = 2m + c
8.5 = 2 (4.25) + c
8.5 – 8.5 = c
We can conclude that the representation of the amount of dog used y, after x weeks in the equation form is:
y = 4.25x
The 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?
The graph shows the relationship between the number of pages printed by a printer and the warm-up time before each printing
We can observe that there is an initial value or y-intercept
The y-intercept from the given graph is: 1
To find the slope from the given graph,
The given points are: (30, 4), and (10, 2)
Slope (m) = \(\frac{2 – 4}{10 – 30}\)
m = \(\frac{-2}{-20}\)
The representation of the equation in the slope-intercept form is:
y = \(\frac{1}{10}\)x + 1
We can conclude that the equation that represents the given situation is:
A function in the form y= mx + b represents a linear relationship between two quantities, x, and y.
?Essential Question How can you use a function to represent a linear relationship?
Make Sense and Persevere 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.
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)
Slope (m) = y2 – y1 / x2 – x1
Compare the given points with (x1, y1), (x2, y2)
–5 = \(\frac{10 – 5}{x – 3}\)
-5 = \(\frac{5}{x – 3}\)
-5 (x – 3) = 5
-5 (x) + 5 (3) = 5
-5x + 15 = 5
-5x = 5 – 15
-5x = -10
x = \(\frac{10}{5}\)
We can conclude that the value of x is: 2
Reasoning What is the initial value of all linear functions that show a proportional relationship?
The representation of the proportional relationship is:
y = mx + 0
m is the slope (or) rate of change
From the above equation,
We can say that the initial value (or) the y-intercept is: 0
We can conclude that the initial value of all linear functions that show a proportional relationship is: 0
Write a function in the form y = mx + b for the line that contains the points (-8.3, -5.2) and (6.4, 9.5).
(-8.3, -5.2) and (6.4, 9.5)
m = \(\frac{9.5 + 5.2}{6.4 + 8.3}\)
m = \(\frac{14.7}{14.7}\)
y = x + c
To find the value of c,
Substitute any one of the points in the above equation
-5.2 = -8.3 + c
c = 8.3 – 5.2
c = 3.1
We can conclude that the representation of the linear equation for the given points is:
y = x + 3.1
The data in the table below represent a linear relationship. Fill in the missing data.
It is given that the data in the table represent a linear relationship
For a linear relationship, for each value of x, there is only 1 value of y
To find the missing data,
Find the rate of change for x and y
The rate of change = 20 – 10
The completed table with the missing data is:
What is an equation that represents the linear function described by the data in Item 5?
From the data in Item 5,
Slope (m) = \(\frac{The rate of change of y}{The rate of change of x}\)
The equation that represents a linear relationship is:
The equation that represents the linear function described by the data in Item 5 is:
A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.
(4, 19) and (9, 24)
m = \(\frac{24 – 19}{9 – 4}\)
19 = 4 + c
c = 19 – 4
y = x + 15
What is a linear function in the form y = mx + b for the line passing through (4.5, -4.25) with y-intercept 2.5?
A line passing through (4.5, -4.25) with y-intercept 2.5
(4.5, -4.25) and (0, 2.5)
m = \(\frac{2.5 + 4.25}{0 – 4.5}\)
m = \(\frac{6.75}{-4.5}\)
m = -1.5
y = -1.5x + 2.5
A 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?
A car moving at a constant speed passes a timing device at t = 0. After 8 seconds, the car has traveled 840 feet
For a constant speed,
Time is considered as input and distance is considered as the output
It is also given that time will start from (0, 0) i..e, at t = 0
The representation of the equation that passes through the origin is:
m is the rate of change (or) slope
In this situation,
The rate of change = \(\frac{Distance}{Time}\)
y = \(\frac{840}{8}\)x
y = 105x
We can conclude that the linear equation that represents the distance traveled by a car after passing the timing device is:
At 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?
At time t = 0, water begins to drip out of a pipe into an empty bucket
Time is considered as output and the amount of water in the bucket is considered as the input
The rate of change = \(\frac{Time}{The amount of water}\)
y = \(\frac{56}{8}\)x
y = 7x
We can conclude that the linear equation that represents the amount of water in inches, w, in the bucket after t minutes
The 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.
The graph of the line represents the cost of renting a kayak
We can observe that the line does not pass through the origin and it has the initial value i..e, the y-intercept
The points from the graph are: (2, 12), and (4, 16)
We can conclude that the relationship to represent the total cost, c, of renting a kayak for t hours is:
An online clothing company sells custom sweatshirts. The company charges $6.50 for each sweatshirt and a flat fee of $3.99 for shipping.
a. 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.
The total cost of the sweatshirts = The cost of each sweatshirt + The flat fee of the sweatshirt for shipping
Let the number of sweatshirts be x
let the total cost of the sweatshirts be y
y = $3.99 + $6.50x
The linear equation representation in the slope-intercept form is:
We can conclude that the equation that represents the total cost, y, in dollars, for a single order of x sweatshirts is:
y = $3.99 + 6.50x
b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.
The equation that represents the total cost, y, in dollars, for a single order of x sweatshirts, is:
$3.99 is the shipping charge for x sweatshirts,
If the shipping charge applied to each sweatshirt, then
The representation of the linear equation which we obtained in part (a) is:
y = $6.50x + \(\frac{$3.99}{x}\)
We can conclude that the linear function that represents if the shipping charge applied to each sweatshirt is:
A store sells packages of comic books with a poster.
a. Model with Math 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.
b. Construct Arguments Suppose another store sells a similar package, modeled by a linear function with initial value $7.99. Which store has the better deal? Explain.
It is given that a store seller sells packages of comic books with a poster
Let x be the number of comics
Let y be the amount obtained by selling comics & poster
For x = 6, y = $12.75
For x = 13, y = $19.75
The representation of the linear equation in the slope-intercept form is:
The points are: (6, 12.75), and (13, 19.75)
m = \(\frac{19.75 – 12.75}{13 – 6}\)
12.75 = 6 + c
c = 12.75 – 6
c = 6.75
We 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:
y = x + 6.75
It is given that another store sells a similar package, modeled by a linear function with an initial value of $7.99
The initial value is: $6.75
From the above initial values,
$6.75 < $7.99
We can conclude that another store has the better deal
Higher-Order Thinking Recommendations for safely thawing a frozen turkey are provided on the packaging.
a. What is the thaw rate of the turkey in hours per pound for refrigerator thawing? For cold water thawing?
b. 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.
The thaw rate is nothing but the rate of change
For refrigerator thawing,
Rate of change = \(\frac{The number of pounds}{Time}\)
= 4 pounds per day
We know that
1 day = 24 hours
Rate of change = \(\frac{4}{24}\)
= \(\frac{1}{6}\) pound per hour
For Cold water thawing,
= 1 pound per 30 minutes
1 hour = 60minutes
Rate of change = \(\frac{1 (2)}{1}\)
= 2 pounds per hour
The rate of change for refrigerator thawing is: \(\frac{1}{6}\) pounds per hour
The rate of change for cold water thawing is: 2 pounds per hour
The representation of the linear equation as a function of weight w in pounds is:
Substitute (24, 4) in the above equation [ The time as x in hours and the weight as y]
4 = \(\frac{24}{6}\) + c
4 = 4 + c
We can conclude that the representation of the linear equation as a function of weight w in pounds is:
Reasoning The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
a. What is the constant rate of change? What does it represent?
The graph shows the relationship between the number of cubic yards of mulch ordered and the total cost of the mulch delivered.
To find the rate of change,
The points are: (20, 450), and (10, 300)
m = \(\frac{300 – 450}{10 – 20}\)
m = \(\frac{150}{10}\)
We can conclude that the constant rate of change for the given graph is: 15
b. What is the initial value? What might that represent?
The initial value is nothing but the y-intercept
We can conclude that the initial value is 50 and this value represents the initial cost of the mulch
An 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?
A. y = x + 2.75
B. y = 3x + 2.75
C. y = x + 3
D. y = 3x + 3
An 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
Let x be the number of samples
Let y be the cost of samples &Admission
For x = 3, y = $5.75
The points are: (3, 5.75), and (6, 8.75)
m = \(\frac{8.75 – 5.75}{6 – 3}\)
5.75 = 3 + c
c = 5.75 – 3
We can conclude that option A matches the given situation
Some eighth-graders are making muffins for a fundraiser. They have already made 200 muffins and figure they can make 40 muffins in an hour.
Write 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.
Let x be the number of hours
The 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
y = 40x + 200
A 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:
How many additional hours would the students spend to make 640 muffins?
It is given that the number of muffins students made is: 640
y = 640
640 = 40x + 200
40x = 640 – 200
40x = 440
x = 11 hours
We can conclude that the additional hours the students would spend to make 640 muffins is: 11 hours
Lesson 3.5 Intervals of Increase and Decrease
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?
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.
The distance is constant for the 2 routes
As speed increases, the time decreases
For route 1:
Route 1 is divided into 2 parts
For the first part (Going up),
As there is friction,
The speed decreases as time increases
For the second part (Going down),
The speed increases as time decreases
For the first part (Going down),
For the second part,
The speed is constant as time increases
Reasoning How do the characteristics of each route affect Martin's travel time and speed?
For the constant distance,
Speed ∝ \(\frac{1}{Time}\)
Route 1 is in the shape of a parabola
In route 1,
For the first part,
Route 2 is increasing for some time and later becomes constant
How does a qualitative graph describe the relationship between quantities?
The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.
The graph at the right shows another interval in the train's travel. Which best describes the behavior of the train in the interval shown?
As time , the speed of the train
The function is
It is given that the graph shows another interval in the train's travel
As speed decreases, the time increases
The function of the given graph is decreasing in nature
How would the graph of the function change if the speed of the train was increasing?
If speed increases, then the time decreases for a constant distance
For a constant distance,
The graph of the function is decreasing in nature as the speed of the train is increased
Write a scenario that the graph above could represent. (Example 3)
The scenario that the given graph could represent is:
The traveling of a vehicle on a hill
You can describe the relationship between two quantities by analyzing the behavior of the function relating the quantities in different intervals on a graph.
Essential Question How does a qualitative graph describe the relationship between quantities?
Look for Relationships How would knowing the slope of a linear function help determine whether a function is increasing or decreasing?
The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right. For a decreasing function, the slope is negative. The output values decrease as the input values increase.
Use Structure What kind of graph of a function shows the same output values, or y-values, for each input value, or x-value?
The vertical line test can be used to determine whether a graph represents a function. A vertical line includes all points with a particular x value. The y value of a point where a vertical line intersects a graph represents an output for that input x value.
What does the graph of the function at each interval represent?
The given graph is divided into 2 intervals
In the first interval,
The height increases with the increase of time
In the second interval,
The height decreases with the increase of time
In which intervals is the function increasing, decreasing, or constant?
The graph is said to be increasing in nature when the line moves from left to right or the slope is positive
The graph is said to be decreasing in nature when the line moves from right to left or the slope is negative
The graph is said to be constant in nature when the line is parallel to any one of the coordinate axes
The intervals that the function is constant – 1, 5
The intervals that the function is increasing – 3, 4
The intervals that the function is decreasing – 2, 6
Use the graph to complete the statements.
The function is in intervals 1, 3, and 6.
The function is in intervals 2 and 5.
The function is constant in interval
The intervals that the function is constant – 4
The intervals that the function is increasing – 1, 3, 6
The 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.
The graph below shows the temperature in Paula's house over time after her mother turned on the air conditioner
As time increases, the temperature decreases
Temperature (°F) ∝ \(\frac{1}{Time}\)
You 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.
In interval (a), the function is
In the interval (b), the function is
In the interval (c), the function is
In interval (d), the function is
You have a device that monitors the voltage across a lamp over time. The results are shown in the graph
In interval (a), the function is increasing
In the Interval (b), the function is constant
In the interval (c), the function is decreasing
In the interval (d), the function is constant
The 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?
The graph below shows the height of a roller coaster over time during a single ride.
The given graph with the intervals is:
The intervals which are increasing in nature are: 1, 3, 5
We can conclude that the increase is the greatest in the 1st interval
Reasoning The graph shows the speed of a car over time. What might the constant intervals in the function represent?
The graph shows the speed of a car over time
The constant lines represent that the speed becomes constant even though the time is increasing
We can conclude that the constant lines in the given graph represent the constant speed
Higher-Order Thinking A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
a. In how many intervals is the function decreasing?
b. How are the decreasing intervals alike?
A signal generator is used to generate signals for a lab experiment over time. The graph shows the frequency of the signal generated.
The given graph with the marked intervals are:
The intervals that the function is decreasing are: 3, 7, 11
The decreasing intervals are all decreasing in nature and have a negative slope
c. How are the decreasing intervals different?
The sizes of the decreasing intervals are different
The values of the negative slopes for decreasing intervals will also be different
Critique Reasoning 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.
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.
The number of intervals that are constant in nature is: 4
We can conclude that the claim of the student is not correct
b. What error might the student have made?
The student considered only the highest constant intervals but not the lowest intervals
The 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
The above are the errors the student has made
Look for Relationships The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
The graph shows the speed of a roller coaster over time. Describe the relationship of speed as a function of time.
First, 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
Which statements about the graph are true? Select all that apply.
The graph is decreasing in intervals (1) and (4).
The graph shows a constant function in interval (2).
The graph is increasing in intervals (2) and (4).
The graph has a constant rate of change.
Let the given options be named as A, B, C, D, and E respectively
We can conclude that the options that match with the given situation are: A and E
Lesson 3.6 Sketch Functions from Verbal Descriptions
Explain It!
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.
A. Do you agree or disagree with Jack's statement? Construct an argument based on the graph to support your position.
The statement of Jack is:
Oil consumption in the United States has dropped a lot
We 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
We can conclude that we can agree with Jack's statement
B. Do you agree or disagree with Ashley's statement? Construct an argument based on the graph to support your position.
The statement of Ashley is:
Oil consumption in China is the biggest problem facing the world environment.
We 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
We can conclude that we can agree with Ashley's statement since China's consumption of oil will make the other countries suffer
Look for Relationships What trend do you see in oil consumption in the United States and in Europe?
The trend in the oil consumption of the United States and Europe (The first line and the second line in the graph) is:
a) The consumption increases gradually for both the United States and Europe from 2000 to 2003
b) From 2004 to 2008, the oil consumption of the United States becomes constant but the consumption of Europe increases gradually
c) From 2009 to 2011, the oil consumption of the United States and Europe decreases gradually
How does the sketch of a graph of a function help describe its behavior?
We can observe that,
At t = 0,
The level of oxygen is full
At t = 22 minutes,,
The level of oxygen decreases
The behavior of the time and the oxygen level is:
Time ∝ \(\frac{1}{Oxygen level}\)
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.
The input, or x-variable, is
The output, or y-variable, is
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.
From the above statements,
As the depth of the water increases, the water pressure also increases
The input or x-variable for the given situation is: Depth
The output or y-variable for the given situation is: Water pressure
To draw the graph,
The required points are: (10, 19.1), and (14, 20.9)
the equation of the straight line in the slope-intercept form is:
m = \(\frac{20.9 – 19.1}{14 – 10}\)
m = \(\frac{1.8}{4}\)
y = \(\frac{9}{20}\)x + c
19.1 = \(\frac{9}{20}\) (10) + c
c = 19.1 – 4.5
c = 14.6
y = \(\frac{9}{20}\)x + 14.6
The representation of the above equation in the coordinate plane is:
Generalize How are the sketches of the two functions similar? How are they different?
From Example 1 and the above item's graph,
We can observe that both graphs have the equation in the slope-intercept form
But,
The graph in Example 1 is decreasing in nature
The graph in the above item is increasing in nature
Haru 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.
Haru 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.
The graph of the relationship of Haru's distance from the house over time is:
You can sketch a graph of a function to describe its behavior. When sketching a function, follow these steps:
1. Identify the two variables (input, output) that have a relationship.
2. Analyze the situation. Look for keywords that indicate that the function is increasing, decreasing, or constant.
3. Sketch the graph.
? Essential Question How does the sketch of a graph of a function help describe its behavior?
Make Sense and Persevere How do you know which variable goes with which axis when you graph?
The independent variable belongs on the x-axis (horizontal line) of the graph and the dependent variable belongs on the y-axis (vertical line).
Reasoning How can you determine the shape of a graph?
The 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 graph with a single peak is called unimodal. A single peak over the center is called bell-shaped. And, a graph with two peaks is called bimodal.
A class plants a tree. Sketch the graph of the height of the tree over time.
a. Identify the two variables.
It is given that a class plants a tree
From the given figure,
As time increases, the height of the tree increases
Input variable (or) x-coordinate: Time
Output variable (or) y-coordinate: Height of the tree
b. How can you describe the relationship between the two variables?
We can conclude that the relationship between the variables is:
Time ∝ Height of the tree
c. Sketch the graph.
The required points are: (0, 3), and (3, 7)
c = 3 – 0
An 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.
An 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.
The graph of the height of the plane over time is (Height of the plane is any value) is:
What relationship between money (in dollars) and time (in months) does this graph show? Write a description of the given graph.
The amount of money is in an increasing trend for the first half of the month
The amount of money is in a decreasing trend for the second half of the month
When 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.
When 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
The graph which represents the given function over the six weeks is:
Aaron'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.
Aaron's mother drives to the gas station and fills up her tank. Then she drives to the market.
At 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
After that, when she drives to the market, the amount of fuel will be in a decreasing trend over time
The graph that shows the relationship between the amount of fuel in the gas tank of her car and time is:
Melody 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.
Melody 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
The graph that represents the above description is:
Which description best represents the graph shown?
A. 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.
B. One train arrives and some people get off the train and wait in the station.
C. People are waiting for a train. Everyone gets on the first train that comes.
D. 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.
At first, there are some people and after some time, the number of people decreased
After some time, there are some people other than the people that decreased and after some further time, the total number of people becomes zero
We can conclude that option D matches the above description
A 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.
A baker has already made 10 cakes. She can make the same number of cakes each hour, which she does for 5 hours.
The initial value for the given graph is: 10
The graph of the relationship between the number of cakes made and time is:
Model with Math 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.
An 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
The projectile (Graph) of an air cannon is always like a "Parabola"
The graph that represents the given situation is:
Higher-Order Thinking 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.
The verbal description of the relationship between the total people and time in hours is:
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.
The graph that describes the above verbal description is:
A 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.
Sketch the graph of this situation below.
A 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.
The graph for the above situation is:
How would the graph change if the innings in which the team scores runs changes?
If the innings in which the team scores runs change, then
The graph will be either in an increasing trend or in a decreasing trend
Topic 3 REVIEW
? Topic Essential Question
How can you use functions to model linear relationships?
Another 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 b is the initial value of the dependent variable.
Vocabulary Review
Match each vocabulary term with its definition.
Each vocabulary term with its definition is:
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)
(0, -2) and (2, 6)
m = \(\frac{6 + 2}{2 – 0}\)
y = 4x + c
-2 = 0 + c
c = -2 – 0
c = -2
Concepts and Skills Review
A 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.
A relation is said to be a function if each input has only 1 output
We can observe that each input has only 1 output
Does the relation {(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)} represent a function? Use the arrow diagram. Then explain your answer.
The given ordered pairs are:
(-5, -3), (7, 2), (3,8), (3, -8), (5, 10)
The ordered pairs are in the form of (input (x), output (y))
The left side represents the values of the input
The right side represents the values of the output
The representation of the ordered pairs in the arrow diagram is:
A relation is said to be a function only when each input corresponds with only 1 output
From the arrow diagram,
The same input corresponds with 2 outputs
You can represent a function in a table, in a graph, or as an equation. The graph of a linear function is a straight line.
Mark has a $100 gift card to buy apps for his smartphone. Each week, he buys one new app for $4.99.
1. Write an equation that relates the amount left on the card, y, over time, x.
Let y be the amount left on the card
The 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
y = $100 – $4.99x
The equation that relates the amount left on the card, y, over time, x is:
2. Make a graph of the function.
You can compare functions in different representations by looking at the properties of functions: the constant rate of change and the initial value.
Two linear functions are shown.
Function A:
c is the initial value (or) y-intercept
The rate of change is: -3
The initial value is: 2
Function B:
The rate of change for all the values of x is: 1
The rate of change for all the values of y is: 2
The rate of change for the table = \(\frac{y}{x}\)
The initial value is the value of y when x = 0
For the given table,
Which function has the greater initial value? Explain.
Function A has the greater initial value
Which function has the greater rate of change?
Function B has the greater rate of change
A 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.
What 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?
A line passing through (0.5, 4.25), and (2, 18.5) and has a y-intercept of -0.5
(0.5, 4.25) and (2, 18.5)
m = \(\frac{18.5 – 4.25}{2 – 0.5}\)
m = \(\frac{14.25}{1.5}\)
m = 9.5
y = 9.5x – 0.5
The graph shows the relationship of the number of gallons being drained from an aquarium over time. What function models the relationship?
The graph shows the relationship of the number of gallons being drained from an aquarium over time.
The y-intercept from the given graph is: 90
The given points are: (0, 90), and (9, 0)
Slope (m) = \(\frac{0 – 90}{9 – 0}\)
m = \(\frac{-90}{9}\)
m = -10
y = -10x + 90
We can conclude that the equation that represents the given relationship is:
LESSON 3-5 Intervals of Increase and Decrease
You 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.
The graph shows the altitude of an airplane over time.
In which intervals is the graph of the function constant? Explain.
The intervals in which the graph of the function is constant are: 2, 4, 6
In which intervals is the graph of the function decreasing? Explain.
The intervals in which the graph of the function is decreasing in nature are: 5, 7
You 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.
Jack'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.
Jack'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
At first,
The number of carrots present in a bowl is: An arbitrary number
The number of carrots Jack consumed is: 0
The graph that shows the number of carrots in the bowl over time is:
What's the Message?
In each row, determine which equation has the greater solution. Circle the letter next to the equation with the greater solution in each row.
The row of equations that has great solutions are:
What does the zero say to the eight?
The statement said by zero to eight is:
You are greater than me