chapter 7 linear functions. explore! relations & functions a relation expresses how objects in...

Chapter 7Linear Functions

EXPLORE! Relations & FunctionsA RELATION expresses how objects in one group (inputs) are

assigned or related to objects in another group (output).

Look at the mapping diagram. It shows three possible relations. If each item, or input, has only one price, or output, then the relation

is also a FUNCTION!

The first two relations are functions because each

item has only one price! The third relation is NOT a

function because the binder has two prices.

Copy the relation diagram. Draw lines from the input

values to the output values so that the relation is a

function. Do this again and draw lines so that the

relation is NOT a function.

Equations and Functions7-1-B

Suppose you can buy magazines for $4 each.

1. Copy and complete the table to find the cost of 2, 3, and 4 magazines

2. Describe the pattern in the table between the cost and the number of magazines.

Equations & Functions

A RELATION is a set of ordered pairs.

A FUNCTION is a relation in which each member of the input is paired with exactly one member of the output.

A FUNCTION RULE is the operation (s) performed on the input value to get the output value.

1. So, what were the ordered pairs in the previous table?

2. Was this a function?3. What was the function rule?


A FUNCTION TABLE is where we are going to

organize the input numbers, the output

numbers, and the function rule.

The set of input values is called the DOMAIN.

The set of output values is called the RANGE.

EXAMPLE!!!Jill saves $20 each month. Make

a function table to show her savings after 1, 2, 3, and 4 months. Then identify the

domain and range.

Domain: {1, 2, 3, 4}Range: {20,40,60,80}


Try it Yourself!A student movie ticket costs $3. Make a function table that shows the total cost for 1, 2, 3, and 4 tickets. Then identify

the domain and range.Input Function Rule Output

Number of Tickets

Cost of Ticket ($)

1

2

3

4

Functions are often written as equations with two

variables to represent the input and output.

Think back to Example 1 & look at this!

Can you write an equation for the movie ticket problem?

Equations and Functions

An armadillo sleeps 19 hours each day. Write an

equation using two variables to show the

relationship between the number of hours h an armadillo sleeps in d

days.

Input Function Rules Output

Number of Days (d)

Number of Hours Slept

(h)

1

2

3

4

h = 19d

ExamplesA botanist

discovers that a certain species of bamboo grows 4

inches each hour. Write an equation

using two variables to show the relationship

between the growth g in inches

of this bamboo plan in h hours.

Melanie read 14 pages of “Hound

of the Baskervilles” each

hour. Write an equation using two variables to

show the relationship between the

number of pages p she read in h

hours.

Anna earns $6.00 an hour working at a grocery story. Make a function table that shows Anna’s total earnings for working 1, 2, 3, and 4 hours. Then identify the

domain and range.

Function TablesCopy and complete each function table. Then, identify the domain

and range.

x0.5x

y

1

2

3

4

x2/3x

y

2

3

4

5

y = 0.5x y = 2/3x

Self Assessment: Complete p. 380 # 1-4 all by yourself. Then check your work with a partner.

More Relations and Functions7-1-B extra

Independent Variables:Values (the Input) that are

chosen and do NOT depend on the other variable.

In an ordered pair, the x-value

Dependent Variables:Values (the Output) that

depend on the Input

In an ordered pair, the y-value


Remember the last section? Determine whether the relation is a function. Explain.

Another way to determine whether a relation is a function is to apply the VERTICAL LINE TEST. 1. Use a pencil to represent a vertical line.2. Place the pencil at the left of the graph3. Move the pencil to the right.4. If for each value of x in the domain, the

pencil passes through only one point on the graph, then the graph represents a function.


A function that is written as an equation can also be written in a form called FUNCTION NOTATION!

y = 4x f(x) = 4xThe function

notation, f(x) is read

“f of x”

The variable y and f (x) both represent the

dependent variable.

1. Find f(3) if f(x) =5x2. Find f(4) if f(x)=2x3. Find f(-5) if f(x)= 20x4. Find f(-8) if f(x)= -5x + 10


Determine whether each relation is a function. Explain.


Determine a rule for each set of ordered pairs. Then copy and complete each function table.

f(x) = x -3 f(x) = 2x +6f(x) = 4x -1

Functions and Graphs

7-1-CThe Westerville Marching Band is going on a year-end trip to an amusement park. Each band member must pay an admission price of $15. In the table, this is represented by 15m. 1. Copy and complete the

function table for the total cost of admission.

2. Graph the ordered pairs (number of members, total cost)

3. Describe how the points appear on the graph.The total cost is a FUNCTION of the number of band members. In general, the output y is a function of the input x.

The graph of the function consists of the points in the coordinate plane that correspond to ALL the ordered pairs of the form (input, output) or (x,y)

Number of

Members

15m Total Cost ($)

1 15(1) 15

2 15(2) 30

3 15(3)

4

5

6

Functions and Graphs

7-1-CThe table shows temperatures in

Celsius and the corresponding temperatures in Fahrenheit. Make a

graph of the data to show the relationship between Celsius and

Fahrenheit.

Celsius (input)

Fahrenheit

(output)

5 41

10 50

15 59

20 68

25 77

30 86

Graph Solutions of Linear Equations

Solution: consists of two numbers; one for each variable; make the equation true. Usually written as an ordered pair

(x,y)

Graph y = 2x +11. Create a table in

which to organize your solutions

2. Select any four values for the input (x)

3. Fill in the table to find the output (y) and then the ordered pairs.

x 2x + 1 y (x,y)

2

1

0

-1

x 2x + 1 y (x,y)

2 2(2) + 1 5 (2,5)

1 2(1) + 1 3 (1,3)

0 2(0) + 1 1 (0,1)

-1 2(-1) + 1 -1 (-1, -1)

Graph Solutions of Linear Equations

Graph each equation.1.y = x – 32.y = -3x + 2

1. Create a table in which to organize your solutions

2. Select any four values for the input (x)

3. Fill in the table to find the output (y) and then the ordered pairs.

x function y (x,y)

Linear FunctionA function like y = 2x + 1 is called a LINEAR FUNCTION because its graph is a line.

x 2x + 1 y (x,y)

2 2(2) + 1 5 (2,5)

1 2(1) + 1 3 (1,3)

0 2(0) + 1 1 (0,1)

-1 2(-1) + 1 -1 (-1, -1)

We already graphed this linear function earlier today. But if you draw a line through the points, you will have graphed ALL of the solutions.

Linear FunctionsMichael Phelps swims the 400-meter individual medley at an average speed of 100 meters per minute. The equation d = 100t describes the distance d that Michael can swim in t minutes. Represent this function

as a graph.

t 100t d (t,d)

1

2

3

4

Examples!

Sandi makes $6 an hour

babysitting. The equation m=6h describes how

much money m she makes

babysitting for h hours. Represent the function by a

graph.

Graph the equation:y= 3x - 1

Graph the function represented by this

table which describes calories in fruit cups:

Servings Calories

1 70

3 210

5 350

7 490


EXPLORE! Rate of Change

Pampered Pets is a doggie daycare where

people drop off their dogs while they work. Mrs. Bybee takes Layla to

Pampered Pets several days a week. The table

shows their prices. Use a graph to determine how the number of hours is

related to the cost.

Number of Hours Cost ($)

1 3.00

2 6.00

3 9.00

4 12.00

5 15.00

6 18.00

Create a

graph of this data!

1. Is the graph linear? Explain2. What is the cost per hour, or unit rate, charged by Pampered Pets?3. Examine any two consecutive ordered pairs from the table. How do the

values change. 4. Is this relationship true for any two consecutive values in the table?5. Use the graph to examine any two consecutive points. By how much

does y change? By how much does x change?6. How does this change relate to unit rate?

Constant Rate of Change

7-2-BThe table shows Andi’s height at

ages 9 and 12.1. What is the change in Andi’s

height from ages 9-12?2. Over what number of years did

it take place?3. Write a RATE that compares the

change in Anei’s height to the change in age. Write this now as a unit rate.

RATE OF CHANGE- a rate that describes how one quantity changes in relation to another. Usually expressed as a unit rate.

CONSTANT RATE OF CHANGE- rate of change in a linear relationship.

Age (years)

Height (inches)

9 53

12 59

Constant Rate of Change

The table shows the amount of money a booster club makes washing

cars for a fundraiser. Use the information to find the constant rate

of change in dollars per car.

The table shows the number of miles a

plane traveled while in flight. Use the

information to find the constant rate of change

in miles per minute.

About 10 miles per minute

Using a Graph to find Rate of Change

The graph represents the distance traveled

while driving on a highway. Use the graph to find the constant rate of

change in MILES per HOUR.

To do this, find any two points on the line such as (1,60) and

(2,120)

Use the graph to find the constant rate of change in miles per hour while driving in

the city.30 mile per

hourSelf Assessment: Complete p. 393 # 1-7 all by yourself. Then check your work with a partner.

Slope7-2-C

In a LINEAR RELATION, the vertical change per unit of horizontal change is always the SAME

We call this change the SLOPE!

Remember CONSTANT RATE OF CHANGE??? Well…it is the same as the slope.

SLOPE basically tells how steep a line is.

Find the SLOPE

The table shows the relationship between the number of seconds y it

takes to hear thunder after a lightening strike and the miles x you are from the

lightening.

Miles (x) Seconds (y)

0 0

1 5

2 10

3 15

4 20

5 25

Find the SlopeGraph the data about plant

height for a science fair project. Then find the

slope of the line. EXPLAIN what the slope represents.

The table below shows the relationship between the number

of hours Carl works and the amount of money that he earns. Graph the data. Find the slope of the line! Then EXPLAIN what the

slope represents.

Week Plant Height

1 1.5

2 3

3 4.5

4 6

Hours Amount Earned ($)

3 45

6 90

9 135

Interpret SlopeRenaldo opened a savings account with the $300 he earned mowing

yards over the summer. Each week he withdraws $20 for expenses.

Draw a graph of the account balance versus time.

Find the numerical value of the slope and interpret it in words.

So, in order to draw the

graph, you’re going to

need a table. The x

values should be the

______ and the y values

should be the ______.

The slope is $20/week. It is actually negative

though b/c his balance declines!


Direct Variation7-3-C

A car travels 130 miles in 2 hours, 195 miles in 3 hours, and 260 miles in 4 hours, as shown.1. What is the constant rate of

change, or slope, of the line?2. Is the distance traveled

always proportional to the driving time?

3. Compare the constant rate of change to the constant ratio.

DIRECT VARIATION: the relationship when two variables have

a constant ratioAlso called DIRECT

PROPORTION The constant ratio itself is called CONSTANT OF VARIATION

Find a Constant RatioThe height of the water as a pool is being filled is shown in

the graph. Determine the rate in inches per minute.

The rate of

change is

going to be

constant

because the

graph forms

a line.

The pool

fills at a rate of

0.4 inches every

minute.

EXAMPLE: Two minutes after a diver enters the water, he has descended 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the

scuba diver descending?

Direct Variation

In DIRECT VARIATION, the CONSTANT RATE OF CHANGE, aka the slope, is assigned a

special variable, k.

Determine Direct Variation

Pizzas cost $8 each plus a $3 delivery

charge. Make a table AND graph to show the cost of 1, 2, 3,

and 4 pizzas. Is there a constant rate? A

direct variation?

Number of Pizzas Cost ($)

1 11

2 19

3 27

4 35

Because there is NO CONSTANT RATE,

there is NO DIRECT VARIATION


Two pounds of cheese cost $8.40. Make a table and graph to show the cost of 1, 2, 3, and 4

pounds of cheese. Is there a constant rate? Is there a direct variation?

A photographer charges a $30

sitting fee and then $6 for each photograph

ordered. Make a table and a graph

to show the cost of 1, 2, 3, and 4

photographs. Is there a constant rate? A direct

variation?

INTERESTING AND IMPORTANT FACT:

Just because there is a line/a linear

relationship, there is not necessarily a direct

variation.

Yes; Yes

Yes; No


Determine whether each linear function is a direct variation. If so, state the constant of variation.

Time, x Wages, y

1 12

2 24

3 36

4 48 Time (h), x

Temperature

Change (°), y

2 4

3 5

4 7

5 11

In Summary…


Inverse Variation7-3-E

1. What is the constant in each rectangle?

2. What happens to the width as the length increases?

3. What happens to the length as the width increases?

This example shows INVERSE VARIATION

INVERSE VARIATION means the product of x and y is a constant. We say that y is inversely

proportional to x.As x increases, the value of y

decreases, but not at a constant rate so the graph of an inverse variation is not a

straight line.

Inverse VariationThe table shows the relationship

between the frequency and wavelength of a musical tone.

Determine if the relationship is an inverse variation. Justify your

response.1. Look at the data.2. Graph the data in the table3. Analyze the graph.

Frequency

(vibrations per

second)

Wavelength (feet)

220 4

440 2

660 1 1/3

880 1The graph shows that

this is an INVERSE VARIATION. The x- and y- coordinates each have a

product of 880.

Inverse Variation

The number of carpenters

needed to frame a house varies

inversely as the number of days

needed to complete the

project. Suppose 5 carpenters can frame a certain

house in 16 days. How many days

will it take 8 carpenters to

frame the house? Assume that they

all work at the same rate.

Here’s how to solve:Use INVERSE VARIATION!

Let x be the number of carpenters. Let y be the number of days.

A crew of 8 carpenters can frame the house in 10 days.

Inverse VariationGraph the data in the table. Then determine

if the relationship is an inverse variation.Burn

Time (H)

Volume (in

cubed)

1 128

2 64

3 32

4 16

The table shows the relationship between the

cost of a gift and the number of friends splitting the cost. Determine if the relationship

is an inverse variation. Justify your response.

Graph the data in the table.

Number of Frien

ds

Cost per

Friend

1 $120

2 $60

3 $40

4 $30

Inverse VariationThe number of

hours it takes to wash windows at an office building varies inversely

as the number of washers. Suppose two washers can get all of them washed in 64

hours. How many hours will it take

8 washers? Assume they all

work at the same rate.

The number of bricklayers needed to build a brick wall

varies inversely as the number of hours needed. Four brick

layers can build a brick wall in 30 hours. How long would it take 5 bricklayers to build a

wall?

24 hours

16 hours


chapter 7 linear functions. explore! relations & functions a relation expresses how objects in...

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