Representing Relationships

Representing RelationshipsChapter 4, Lesson 1

Real-World Link To achieve orbit, the space shuttle must travel at a rate of about 5 miles per second. The table shows the total distance d that the craft covers in certain periods of time t.a. Write an algebraic expression for the

distance in miles for any number of seconds t.

5t

b. Describe the relationship in words. The distance is 5 times the number of

seconds.

Real-World Link To achieve orbit, the space shuttle must travel at a rate of about 5 miles per second. The table shows the total distance d that the craft covers in certain periods of time t.c. Graph the ordered pairs. Describe the shape of the graph.

Linear EquationAn equation with a graph that makes a straight line.

Can have more than one variable.

Example 1Write an equation to find the number of liters in any number of quarts. Describe the relationship in words.

The rate of change is 0.95.

EQUATION: = 0.95q

IN WORDS:The rate of change between quarts and liters is 0.95.

Example 2About how many liters are in 8 quarts?

EQUATION: = 0.95q = 0.95(8) = 7.6

There are about 7.6 liters in 8 quarts.

Got it? 1 & 2The total cost of tickets to the school play is shown in the table. a. Write an equation to find the total cost of any number of tickets. Describe the relationship in words. EQUATION: c = 4.5tWORDS: Each ticket cost $4.50.b. Use the equation to find the cost of 15 tickets.

c = 4.5tc = 4.5(15)c = 67.50

The cost of 15 tickets is $67.50.

Example 3The total distance Marlon ran one week is shown by the graph. Write an equation to find the number of miles run y after any number of days x.1. Find the rate of change.

m = =

2. Find the y-intercept. y = mx + b

y = 3.5x + b 7 = 3.5(2) + b 0 = b

The slope is 3.5 and the y-

intercept is 0. The equation is

y = 3.5x.

Example 4Using the same equation in Example3, how many miles will Marion run after 2 weeks?

y = 3.5xLet x by 14, since x is in days.

y = 3.5(14)y = 49

Marion will run 49 miles in 2 weeks.

Got it? The number of trees saved by recycling paper is shown. a. Write an equation to

find the total number of trees y that can be saved for any number of tons of paper x.

y = 17x

b. Use the equation to find how many trees could be saved if 500 tons of paper are recycled.

8,500 trees

Multiple Representations of Linear Equations

Words: The number of trees is equal to 17 times the number of tons of paper.

Equation: y = 17x

Table:

x y

1 17

2 34

3 51

4 68

Graph:

Example 5Chloe completes in jump rope competitions. Her average rate is 225 jumps per minute.a. Write an equation to find the number of

jumps in any of amount of minutes. j = 225m

b. Make a table with 1, 2, 3, 4, and 5 minutes. Graph the points.

m 225m

j

1 225(1)

225

2 225(2)

450

3 225(3)

675

4 225(4)

900

5 225(5)

1,125

Got it? Paul earns $7.50 an hour working at a grocery store. a. Write an equation the find the amount of

money Paul earned m for any number of hours h.

b. Make a table to find the earnings if he works 5, 6, 7, and 8 hours. Graph the coordinate points.

RelationsChapter 4, Lesson 2

Relations: a set of ordered pairs

Example 1Express the relation {(2, 6), (-4, 8), (-3, 6), (0, -4)} as a table and a graph. State the domain and range.

Domain:{-4, -3, 0, 2}

Range: {-8, -4, 6)

Got it? Express the relation {(-5, 2), (3, -1), (6, 2), (1, 7)} as a table and a graph. State the domain and range.

Domain:{-4, -3, 0, 2}

Range: {-8, -4, 6)

Example 2It cost $3 per hour to park at the Wild Wood Amusement Park. Make a table using x and y coordinates that represent the total cost for 3, 4, 5, and 6 hours.

Graph the ordered pairs.

Got it? 2A movie rental store charges $3.95 per movie rental. Make a table using x and y coordinates for the total cost of 1,2, 3, and 4 movies.


Warm-Up

Are these relations functions? Why or why not?

1. {(7, -4), (8, 2), (8, 1), (-4, 5)}

2. {(9, 3), (2, 3), (4, -3), (7, 2)}

FunctionsChapter 4, Lesson 3

VocabularyFunction:Where every domain (input) is matched up with exactly one range (output)

Example:m = 14p

If m represent the amount of money you earn, and p is the number of pizza’s you deliver.

How much money will you make? Depends on how many pizza’s you deliver.

m = dependent p = independent

Independent Variable

EquationDependent

Variable

number of downloads

The equation c = 0.99n represents the total cost c for n music

downloads.

cost

The equation d = 4.5h represents the number of miles Amber can run

in h hours.

The equation s = g + 3 represents the final

score of games s after g goals in the final

period.

number of hours

number of mile

number of goals

final score

Functions

Example 1:Find f(-3) if f(x) = 2x + 1.

f(x) = 2x + 1f(-3) = 2(-3) + 1

f(-3) = -6 + 1

So, f(-3) = -5.

Function Tables:A way to organize the domain, range and rule on a table.

Independent Variable = DomainDependent Variable = Range

Example 2:Choose four values for x and make a function table for f(x) = x + 5. Then state the domain and range.

Domain is {-2, -1, 0, 1}Range is {3, 4, 5, 6}

Got it? 1 & 2Choose four values for x to complete the function table for the function f(x) = x – 7. Then state the domain and range.

Domain: {-1, 0, 1, 2}Range: {-8, -7, -6, -5}

Example 3There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j.

Identify the independent and dependent variable.

Ask: How many peanuts p(j) are there? It depends…

So, p(j) or the number of peanuts are the dependent variable.

Logic tells us that the number of jars is the independent variable.

Example 4There are approximately 770 peanuts in a jar of peanut butter. The total number of peanuts p(j) is a function of the number of jars of peanut butter j. What values of the domain and range make sense for this situation? DOMAIN:Ask: positive or negative numbers, whole, decimals?

only positive whole numbersRANGE:The range depends on the x-values, and since there are 770 peanuts in each jar, the range will be multiples of 770.


Write a function to represent the total number of peanuts.

p(j) = 770j


How many peanuts are there in 7 jars of peanut butter?

p(j) = 770jp(7) = 770(7)p(7) = 5,390

There will be 5,390 peanuts in 7 jars of peanut butter.

The domain is the set of numbers for the independent variable.

The range is the set of numbers for the dependent variable.

Got it? 3-5A scrapbooking store is selling rubber stamps for $4.95 each. The total sales f(n) is a function of the number of rubber stamps n sold. a. Identify the independent and dependent

variable.independent = n dependent = f(n)

b. What values of the domain and range make sense for this situation?

Domain: positive whole numbersRange: multiples of 4.95

c. Write a function equation to represent total sales.

f(n) = 4.95n

d. Determine the total cost of 5 stamps. $24.95

Linear FunctionsChapter 4, Lesson 4

Sometimes functions are written with two variables, x and y.

x represents the domain

y represents the range

Example 1The school stores buys book covers for $2 each and notebooks for $1. Toni has $5 to spend. The function y = 5 – 2x represents this situation. Graph the function and interpret the points graphed.

Chose values for x and substitute them to find y.


Toni has 4 options at the book store.5 notebooks, 1 cover and 3 notebooks, or 2 covers and 1 notebook.

Got it? 1The farmer’s market sells apples for $2 per pound and pears for $1 per pound. Mallory has $10 to spend. The function y = 10 – 2x represents this situation. Graph this function and interpret the points.

Mallory can purchase10 pounds of pears, or 8pounds of pear and 1pound of apples, or 2pounds of pears and 2pounds of apples.

Example 2Graph y = x + 2.

Make a function table.


Graph these functions.

a. y = x – 5

b. y = -2x

Got it? 2

Representing Functions

Equation: y = x – 1

Table:

Words: the value of y is one less than the corresponding value of x.Graph:

Linear Functions: a function where the graph is a line.

Example: y = mx + b

Continuous vs. discrete dataContinuous – no space between data valuesDiscrete – have space between data values

a. Write a function to represent the total value of coupons given out.

b. Make a function table for 5, 10, 15, and 20 and graph the points.

c. Is the function continuous or discrete? Explain.

Example 3

x 5x y

5 5(5) 25

10 5(10) 50

15 5(15) 75

20 5(20) 100

Each person that enters the store receives a coupon for $5 off his or her entire purchase.

y = 5x

There can only be a whole number of customers, so the graph is discrete.

a. Write a function to represent the total cost of any number of pounds.

b. Make a function table for 1, 2, 3, 4, and 5 pounds and graph the points.

c. Is the function continuous or discrete? Explain.

Got it? 3

x 5.95x y

1 5.95(1) 5.95

2 5.95(2)11.9

0

3 5.95(3)17.8

5

4 5.95(4)23.8

0

5 5.95(5)29.7

5

A store sells trail mix for $5.95 per pound.

y = 5.95x

There can be decimals of pounds and cost so the function is continuous.

Make ordered pairs from the x-value and y-value. Then graph the coordinates and draw a line through, IF the function is continuous.

Warm-upUse a map diagram to show if these relations are function.

a. {(1, -3), (3, 8), (2, 8), (-7, 1)}

b. {(-4, 0), (7, -2), (-4, 5), (2, 9)}

Compare Properties of FunctionsChapter 4, Lesson 5

Carlos and Stephanie are members to the science museum. Carlos’s members can be represented by the function c = 9.99. The cost of Stephanie’s membership is shown by the table.

Real-World Link

a. Make a table to represent the cost of Carlos’s membership.

b. Describe the rate of change for each function.

Carlos has a rate of 0, and Stephanie has a rate of 5.

Months

Cost ($)

1 5

2 10

3 15

4 20

Months

Cost ($)

1 9.99

2 9.99

3 9.99

4 9.99

A zebra’s main predator is a lion. Lions can run at a speed of 53 feet per second over short distances. The graph shows the speed of a zebra. Compare their speeds.

Example 1

Lion’s rate of change = 53

Find the zebra’s rate of change

The zebra has a faster rate than the lion.

A 2013 Ford Acura has a gas mileage of 21 miles per gallon. The gas mileage of a 2013 Audi is represented by this graph. Compare their gas mileage.

Got it? 1

The Ford Acura has a better gas mileage.

The Acura has 21, and the Audi has a rate of 19.

The function m = 140h, where m is the miles traveled in h hours, represents the speed of the first Japanese high speed train. The speed today’s high speed train in China is shown by the table.

Example 2

a. Compare the functions’ y-intercepts and rate of change. They both have the same y-

intercept: 0

The rate of change for Japan is 140.

The rate of change forChina is 217.

China’s high speed train is faster than

Japan’s train.

The function m = 140h, where m is the miles traveled in h hours, represents the speed of the first Japanese high speed train. The speed today’s high speed train in China is shown by the table.

Example 2

b. If you ride each train for 5 hours, how far will you travel on each?

Japan: y = 140hy = 140(5)

y = 700

You will travel 700 miles on Japan’s train. You will travel

1,085 miles in 5 hours on the

Chinese train.

The number of new movies Movie Madness receives can be represented by the function m = 7w + 2, where m represents the number of movies and w represents the number of weeks. The number of games Game Gallery receives is shown in the table.

Got it? 2

a. Compare the functions’ y-intercept and rate of change. Movie Madness has a rate of 7

and a y-intercept of 2.

Game Gallery has a rate of 3 and a y-intercept of 0.

b. How many new movies will each store have in Week 6?

Movie Madness will have 44 new movies and Game Gallery will have 18 new games.

Angela and Ben each have a monthly cell phone bill. Angela’s monthly bill is represented by y = 0.15x + 49, where x represents the amount of minutes and y represents the cost. Ben’s monthly cost is shown by the graph. a. Compare the y-intercepts and rate of change.

Example 3

Angela’s y-intercept is 49 and the rate of change is

0.15.

Ben’s y-intercept is 60.Rate of change =

So, Angela pays more per minute.

Angela and Ben each have a monthly cell phone bill. Angela’s monthly bill is represented by y = 0.15x + 49, where x represents the amount of minutes and y represents the cost. Ben’s monthly cost is shown by the graph. b. What will be the monthly cost for Angela and Ben for 200 minutes?

Example 3

Angelay = 0.15(200) + 49

y = 79Angela will pay $79 for

200 minutes.

According to the graph, Ben will pay $80 for 200

minutes.

Mandy and Sarah each have a membership to the gym. Mandy’s membership is represented by the function y = 3x + 29, where x represents the hours with a trainer and y represents the cost. The cost of Sarah’s membership is shown in the graph.

a. Compare the y-intercepts and rates of change.b. What will be the total cost for Mandy and Sarah if they each have 4 hours with a trainer?

Got it? 3

a. Mandy: y-intercept is 29

Rate of change is $3

Sarah: y-intercept is 39Rate of change is $4

b. Mandy = $41Sarah = $ 51

Lorena’s mother needs to rent a truck to move some furniture. The cost to rent a truck from two different companies are shown with a graph and table. Which company should she use if she wants to rent the truck for 40 miles?

Example 4

Ron’s Rentals will charge $100 for 40 miles.

Cross Town Movers:slope = 0.5 and y-intercept is 30

y = 0.5x + 30

It will cost $70 for 40 miles.

Construct FunctionsChapter 4, Lesson 6

Dylan is planning a birthday party at a skating rink. The rink charges a party fee plus an additional charge for each guest.

Real-life Link

a. Choose two points and find the rate of change.

(3, 59) and (5, 65)The rate of change is 3.

b. Write a function to this situation.

y = 3x + 50

c. Graph the ordered pairs and extend the line until you reach the y-axis. How much is the party fee?

The party fee is $50.

A shoe store offers free points when you sign up for their rewards card. Then, for each pair of shoes purchased, you earn an additional number of points. The graph shows the total points earned for several pairs of shoes. Find and interpret the rate of change.

Example 1

You will earn 15 points for every pair of shoes

you buy.

Meyer Music charges a yearly subscription fee plus a monthly fee. The total cost for different number of months, including the yearly fee, is shown by the graph. Find and interpret the rate of change and the initial value.

Got it? 1

The monthly fee is $3.

The initial value is 15-3, or $12.

Joan has some photos in her photo album. Each week she plans to add 12 photos. Joan had 120 photos after 8 weeks. Assume the relationship is linear. Find and interpret the rate of change and initial value.

The phrase “each week she adds 12” means the rate of change is 12.

One of the points on the line is (8, 120)

Example 2

y = mx + b120 = 12(8) + b

120 = 96 + b24 = b

The y-intercept is 24. So, the initial amount of photos is 24.

A zoo charges a rental fee plus $2 per hour for strollers. The total cost of 5 hours is $13. Assume the relationship is linear. Find the interpret the rate of change and initial value.

The hourly rate is $2, and the rental fee (initial value) is $3.

Got it? 2

The table shows how much money Ava has saved. Assume the relationship is linear. Find and interpret the rate of change and initial value.

Example 3

1. Choose two points to find the rate of change.

Ava saves $20 each month.

2. Use slope-intercept form to find the initial value.

y = mx + b110 = 20(3) + b

50 = bAva initially saved $50.

The table shows the monthly cost of sending text messages. Assume the relationship is linear. Find and interpret the rate of change and initial value.

Got it?

# of Messages

(x)

$ Cost (y)

5 10.50

6 10.60

7 10.70

Each text costs $0.10. the initial cost of the phone plan is $10.

Warm-up Find the initial value and the rate of change for this table.

Is this relation a function?

x y

15 7

20 8.25

25 9.5

30 10.75

m = .25

b = 3.25

Yes. Each x value goes

with 1 y value.

Linear and Nonlinear FunctionsChapter 4, Lesson 7

The table shows the approximate height and horizontal distance traveled by a football kicked at an angle of 30 with an initial velocity of 30 yards per second.

Real-World Link

a. Did the football travel the same height each half-second?

No

b. Did the football travel the same length each half-second?

Yes

The table shows the approximate height and horizontal distance traveled by a football kicked at an angle of 30 with an initial velocity of 30 yards per second.

Real-World Link

c. Graph the ordered pairs (time, height) and (time, length).

Determine if each table represents a linear or nonlinear function. Explain.a.

Example 1

As x increases by 2, y decreases by 15 each time. The table is predictable, so the function is linear.

Determine if each table represents a linear or nonlinear function. Explain.b.

Example 1

As x increases by 3, y increases by different amounts each time. The rate of change is not constant. The function in nonlinear.

Determine whether each table represents a linear or nonlinear function. Explain.

a.

b.

Got it? 1

Linear; as x increases by 5, y decreases by 4.

Nonlinear; as x increases by 2, y

increases by a different

number every time.

Use the table to determine whether the minimum number of Calories a tiger cub should eat is linear function of its age in weeks.

Example 2

Find the rates.

1000 – 825 = 1751185 – 1000 = 1851320 – 1185 = 1351420 – 1320 = 100

The rates of change are not consistent. The function in nonlinear.

Tickets at a school dance cost $5 per student. Are the ticket sales a linear or nonlinear function of the number of tickets sold. Explain.

Got it? 2

Yes, the rate of change is constant; as the number of tickets sold increases by 1, the total ticket sales increases by $5.

A square has a side length of s inches. The area of the square is a function of the side length. Does this situation represents a linear or nonlinear function. Explain.

Make a table to show the area of the square for side lengths of 1, 2, 3, 4, and 5

inches.

Example 3

A square has a side length of s inches. The area of the square is a function of the side length. Does this situation represents a linear or nonlinear function. Explain.

Graph the function.

Example 3

A square has a side length of s inches. The perimeter of the square is a function of the side length. Does this situation represent a linear or nonlinear function. Explain.

Linear; if you graphed the ordered pairs (side length, perimeter) the points would

make a line.

Got it? 3

Quadratic FunctionsChapter 4, Lesson 8

A special type of nonlinear function is a quadratic function.

The greatest power on the variable is 2.Examples:

y = x2 + 3x – 5 y = 2x2 + 2x + 1

The graph is U-shaped, opening upward or downward.

A function in which the greatest power of the

variable is 2.

The graph of a quadratic function sometimes open

upward.

The graph always opens downward.

The graph is a straight line.

Quadratic Functions

TRUE

TRUE

FALSE

FALSE

Graph y = x2.

Example 1

Graph y = -x2 + 4.

Example 2

Graph y = 6x2.

Got it? 1&2

The function d = 4t2 represents the distance d in feet that a race car will travel over t seconds with a constant acceleration of 8 feet per second. Graph the function. Use the graph to find how much time it will take for the race car to travel 200 feet.

Example 3

The function h = 0.66d2 represents the distance d in miles you can see from height of h feet. Graph this function. Then use the graph to estimate how far you can see from a hot air balloon 1,000 feet in the air.

Distance cannot be negative, so use only positive numbers.

Example 4

Warm-up

Which of the following are examples of a nonlinear function?

a. y = 6x + 4 b. 2x2 + y = 8

c. y = 0.6x3 d. 2x + 3y = 4

d. (y – 8) = -4(x2 +1) e. 8y = 6 – 2x

Qualitative GraphsChapter 4, Lesson 9

Emily is downloading photos from her digital camera to her computer. The table shows the percent of photos downloaded for several seconds.

Real-World Link

a. During which period(s) of time did the percent downloaded not change?

between 4 and 6 seconds and between 8

and 10 seconds

b. During which period of time did the percent downloaded change the most?

between 6 and 8 seconds

Emily is downloading photos from her digital camera to her computer. The table shows the percent of photos downloaded for several seconds.

Real-World Link

c. Graph and connect the ordered pairs.

Qualitative Graphs: Graphs used to represent situations that may not have numbers.

The graph displays the water level in a kiddy pool that has a drain. Describe the change in the water level over time.

Example 1

The water level increased at a constant rate, then the water was turned off. After some time, the pool is drained at a constant rate until the water is gone.

The graph displays the revenue from a local clothing store. Describe the sales over time.

Got it? 1

Overall, the sales increase steadily. There are two periods of time where the sales decrease or remain constant.

A tennis ball is dropped onto the floor. On each successive bounce, it rebounds to a height less than its previous bounce height until it comes to rest on the floor. Sketch a qualitative graph.

Draw the axis and label.

Sketch the situation.

Example 2

You swing on a swing. Sketch a qualitative graph to represent the situation.

Draw and label the axes.

Sketch the shape.

Example 3

A car is traveling at a constant speed. The car slows down steadily to come to rest at a stop light. Sketch a qualitative graph to represent the situation.

Got it? 3&4

Documents

Representing Relationships