Determine if each relationship represents a function.

1.

2. y = 3x2 – 1

3. For the function f(x) = x2 + 2, find f(0), f(3),

and f(–2).

yes

yes

2, 11, 6

Warm Up

Pre-Algebra

Linear Functions

12-5

Learn to identify linear functions.

linear function

Vocabulary

The graph of a linear function is a line. The linear function f(x) = mx + b has a slope of m and a y-intercept of b. You can use the equation f(x) = mx + b to write the equation of a linear function from a graph or table.

Write the rule for the linear function.

Use the equation f(x) = mx + b. To find b, identify the y-intercept from the graph.

b = 2

f(x) = mx + 2Locate another point on the graph, such as (1, 4). Substitute the x- and y-values of the point into the equation, and solve for m.

Example: Writing the Equation for a Linear Function from a Graph

f(x) = mx + 2

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

4 = m + 2– 2 – 2

2 = m

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

Example Continued

Write the rule for the linear function.

x

y

2

2-2

4

4-4

-4

Use the equation f(x) = mx + b. To find b, identify the y-intercept from the graph.

b = 1

f(x) = mx + 1Locate another point on the graph, such as (5, 2). Substitute the x- and y-values of the point into the equation, and solve for m.

-2

Try This

f(x) = mx + 1

2 = m(5) + 1 (x, y) = (5, 2)

2 = 5m + 1– 1 – 1

1 = 5m15m =

The rule is f(x) = x + 1.15

Try This Continued

Write the rule for the linear function.

A. The y-intercept can be identified from the table as b = f(0) = 1. Substitute the x- and y-values of the point (1, –1) into the equation f(x) = mx + 1, and solve for m.f(x) = mx + 1

–1 = m(1) + 1

x y

–2 5

–1 3

0 1

1 –1

–2 = m

The rule isf(x) = –2x + 1.

–1 = m + 1–1 –1

Example: Writing the Equation for a Linear Function from a Table

Write the rule for the linear function.

B. Use two points, such as (1, 4) and (3, 10), to find the slope.

Substitute the x- and y-values of the point (1, 4) into f(x) = 3x + b, and solve for b.

x y

–3 –8

–1 –2

1 4

3 10

m = = = = 3 y2 – y1x2 – x1

10 - 43 - 1

62

Example: Writing the Equation for a Linear Function from a Table

f(x) = 3x + b

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

4 = 3 + b

–3 –3

1 = b

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

Example Continued

Write the rule for the linear function.

A. The y-intercept can be identified from the table as b = f(0) = 0. Substitute the x- and y-values of the point (1, –1) into the equation f(x) = mx + 0, and solve for m.

f(x) = mx + 0

–1 = m(1) + 0

x y

0 0

–1 1

1 –1

2 –2

–1 = m

The rule is f(x) = –x.

Try This

Write the rule for each linear function.

B. Use two points, such as (0, 5) and (1, 6), to find the slope.

Substitute the x- and y-values of the point (0, 5) into f(x) = 1x + b, and solve for b.

x y

0 5

1 6

2 7

–1 4

m = = = = 1 y2 – y1x2 – x1

6 – 51 – 0

11

Try This

f(x) = mx + b

5 = 1(0) + b (x, y) = (0, 5)

5 = b

The rule is f(x) = x + 5.

Try This Continued

A video club cost $15 to join. Each video 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) = mx + 15

16.5 = m(1) + 15

The y-intercept is the cost to join, $15.

With 1 rental the cost will be $16.50.16.5 = m + 15–15 – 15

1.5 = m

The rule for the function is f(x) = 1.5x + 15. After 12 video rentals, the cost will be f(12) = 1.5(12) + 15 = 18 + 15 = $33.

Example: Money Application

A 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) = mx + 20

22 = m(1) + 20The y-intercept is the cost to join, $20.

With 1 book purchase the cost will be $22.22 = m + 20–20 – 20

2 = m

The rule for the function is f(x) = 2x + 20. After 10 book purchases, the cost will be f(10) = 2(10) + 20 = 20 + 20 = $40.

Try This

Write the rule for each linear function.

1.

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. Write a function for Andre’s expenses for the day. Determine his expenses if he sold 25 toys.

f(x) = 3x – 1

f(x) = –3x + 2

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

x –2 –1 0 1 2

y 8 5 2 –1 –4

x –3 0 3 5 7

y –10 –1 8 14 20

Lesson Quiz