Linear Functions8-4

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

Linear Functions8-4

Warm UpDetermine if each relationship represents a function.

1.

2. y = 3x2 – 1

3. For the function y = x2 + 2, find when

x = 0, x = 3, and x = –2.

yes

yes

2, 11, 6

Linear Functions8-4

Problem of the Day

Take the first 20 terms of the geometric sequence 1, 2, 4, 8, 16, 32, . . . .Why can’t you put those 20 numbers into two groups such that each group has the same sum?

All the numbers except 1 are even, so the sum of the 20 numbers is odd and cannot be divided into two equal integer sums.

Linear Functions8-4

MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem.Also MA.8.A.1.1

Sunshine State Standards

Linear Functions8-4

Vocabulary

linear function

function notation

Linear Functions8-4

A linear function is a function that can be described by a linear equation. One way to write a linear function is by using function notation. If x represents the input value, then the and y represents the output value, the function notation for y is f(x), where f names the function.

Any linear function can be written in slope-intercept form f(x) = mx +b where m is the slope of the function’s graph and b is the y-intercept.

Linear Functions8-4

Determine whether the function f(x) = 2x3 is linear. If so, give the slope and y-intercept of the function’s graph.

Additional Example 1A: Identifying Linear Functions

The function is not linear because x has an exponent other than 1.

The function cannot be written in the form f(x) = mx + b.

Linear Functions8-4

Determine whether the function f(x) = 3x + 3x + 3 is linear. If so, give the slope and y-intercept of the function’s graph.

Additional Example 1B: Identifying Linear Functions

f(x) = 3x +3x + 3

The function is linear because it can be written in the form f(x) = mx + b. The slope m is 6, and the y-intercept b is 3.

Write the equation in slope-intercept form.

f(x) = 6x + 3 Combine like terms.

Linear Functions8-4

Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.

Check It Out: Example 1A

f(x) = –2x + 4

m = –2; b = 4; f(x) = –2x + 4 is a linear function because it can be written in the form f(x) = mx + b.

Linear Functions8-4

Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.

Check It Out: Example 1B

f(x) =– + 41x

f(x) =– + 4 is not a linear function because

x appears in a denominator.

1x

Linear Functions8-4

Additional Example 2A: Writing the Equation for a Linear Function

Step 1: Identify the y-intercept b from the graph.

b = 2

Step 2: Locate another point on the graph, such as (1, 4).

Step 3: Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for m.

Write a rule for the linear function.

Linear Functions8-4

Additional Example 2A Continued

f(x) = mx + b

4 = m(1) + 2 (x, y) = (1, 4)

4 = m + 2– 2 – 2

2 = m

The rule is f(x) = 2x + 2.

Linear Functions8-4

Additional Example 2B: Writing the Equation for a Linear Function

Step 1: Locate two points.

(1, 4) and (3, 10)

Step 2: Find the slope m.

Step 3: Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.

Write a rule for the linear function.

x y

–3 –8

–1 –2

1 4

3 10m = = = = 3

y2 – y1x2 – x1

10 – 43 – 1

62

Linear Functions8-4

Additional Example 2B Continued

f(x) = mx + b

4 = 3(1) + b (x, y) = (1, 4)

4 = 3 + b– 3 – 3

1 = b

The rule is f(x) = 3x + 1.

Linear Functions8-4

Check It Out: Example 2A

Write a rule for each linear function.

Linear Functions8-4

Check It Out: Example 2A Continued

b = 1; (5, 2): 2 = m(5) + 1

= m; f(x) = x + 115

15

Linear Functions8-4

Check It Out: Example 2B

b = –1; f(x) = 2x – 1

Write a rule for the linear function.

m = = 2 1 –(–1)1 – 0

x –2 –1 0 1 2

y –5 –3 –1 1 3

(0, –1) and (1, 1);

Linear Functions8-4

Example 3: Money ApplicationA video game club cost $15 to join. Each game that is rented costs $1.50. Find a rule for the linear function that describes the total cost of renting videos as a member of the club, and find the total cost of renting 12 videos.

f(x) = 1.5x + 15

f(x) = 1.5(12) + 15

f(x) is the cost of renting games, and x is the number of games rented.

f(x) = 18 + 15

= 33

To write the rule, determine the slope and y-intercept.

m = 1.5b = 15

The rate of change is $1.50 per game.The cost to join is $15.

To rent 12 games as a member will cost $33.

Linear Functions8-4

Check It Out: Example 3A book club has a membership fee of $20. Each book purchased costs $2. Find a rule for the linear function that describes the total cost of buying books as a member of the club, and find the total cost of buying 10 books.

f(x) = 2x + 20

f(10) = 2(10) + 20

= 20 + 20

= 40

rate of change = $2 per book; y-intercept is $20 membership fee;

The total cost of buying 10 books is $40.

Linear Functions8-4

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

Linear Functions8-4

Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.

1. f(x) = 4x2

2. f(x) = 3(x + 4)

Write the rule for the linear function.

3.

Lesson Quiz: Part I

linear; m = 3; b = 12

not linear

1 2

f(x) = x 1

Linear Functions8-4

Write the rule for each linear function.

2.

3. Andre sells toys at the craft fair. He pays $60 to rent the booth. Materials for his toys are $4.50 per toy. Find a rule for the linear function that describes Andre's expenses for the day. Determine his expenses if he made 25 toys.

Lesson Quiz: Part II

f(x) = 3x – 1

f(x) = 4.50x + 60; $172.50

x –3 0 3 5 7

y –10 –1 8 14 20

Linear Functions8-4

1. Identify a function that is linear.

A. f(x) = 4x2

B. f(x) = 2(x2 + 1)

C. f(x) = 2(x + x)

D. f(x) = x2

Lesson Quiz for Student Response Systems

Linear Functions8-4

2. Identify a function that is not linear.

A. f(x) = x

B. f(x) = 0.5x

C. f(x) = 3(x + x) + 2

D. f(x) = 5x2

Lesson Quiz for Student Response Systems

Linear Functions8-4

3. Write the rule for the linear function.

A. f(x) = x + 3

B. f(x) = –x + 3

C. f(x) = x + 3

D. f(x) = 3x + 3

Lesson Quiz for Student Response Systems