Linear Functions
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Linear Functions. 12-5. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Linear Functions. 12-5. Course 3. Warm Up Determine if each relationship represents a function. 1. 2. y = 3 x 2 – 1 3. For the function f ( x ) = x 2 + 2, find f (0), f (3), and f (–2). yes.

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12 5

Linear Functions

12-5

Course 3

Warm Up

Problem of the Day

Lesson Presentation


12 5

Linear Functions

12-5

Course 3

Warm Up

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


12 5

Linear Functions

12-5

Course 3

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.


12 5

Linear Functions

12-5

Course 3

Learn to identify linear functions.


12 5

Linear Functions

12-5

Course 3

Insert Lesson Title Here

Vocabulary

linear function


12 5

Linear Functions

12-5

Course 3

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.


12 5

Linear Functions

12-5

Course 3

Additional Example 1: Writing the Equation for a Linear Function from a Graph

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 + 2

Locate 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.


12 5

Linear Functions

12-5

Course 3

Additional Example 1 Continued

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.


12 5

Linear Functions

12-5

y

4

2

x

-4

-2

2

4

-4

Course 3

Try This: Example 1

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 = 1

f(x) = mx + 1

Locate 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


12 5

Linear Functions

12-5

1

1

5

5

m =

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

Course 3

Try This: Example 1 Continued

f(x) = mx + 1

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

2 = 5m + 1

– 1 – 1

1 = 5m


12 5

Linear Functions

12-5

Course 3

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

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

–1 = m + 1

–1 –1

The rule is

f(x) = –2x + 1.

–2 = m


12 5

Linear Functions

12-5

y2 – y1

m = = = = 3

x2 – x1

10 - 4

3 - 1

6

2

Course 3

Additional Example 2B: 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.


12 5

Linear Functions

12-5

Course 3

Additional Example 2B Continued

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.


12 5

Linear Functions

12-5

Course 3

Try This: Example 2A

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

–1 = m

The rule is f(x) = –x.


12 5

Linear Functions

12-5

y2 – y1

m = = = = 1

x2 – x1

6 – 5

1 – 0

1

1

Course 3

Try This: Example 2B

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.


12 5

Linear Functions

12-5

Course 3

Try This: Example 2 Continued

f(x) = mx + b

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

5 = b

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


12 5

Linear Functions

12-5

Course 3

Example 3: Money Application

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

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

16.5 = m(1) + 15

With 1 rental the cost will be $16.50.

16.5 = m + 15

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.

–15– 15

1.5 = m


12 5

Linear Functions

12-5

Course 3

Try This: Example 3

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

The y-intercept is the cost to join, $20.

With 1 book purchase the cost will be $22.

22 = m(1) + 20

22 = m + 20

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.

–20– 20

2 = m


12 5

Linear Functions

12-5

Course 3

Insert Lesson Title Here

Lesson Quiz

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 + 2

f(x) = 3x – 1

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


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