1. Linear Functions 12-5 Warm Up Problem of the Day Lesson Presentation
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Linear Functions 12-5 Warm Up
Course 3
12-5
Linear Functions
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

3. Linear Functions 12-5 Problem of the Day
Course 3
12-5
Linear Functions
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.

4. Course 3
12-5
Linear Functions
Learn to identify linear functions.

5. Insert Lesson Title Here
Course 3
12-5
Linear Functions
Insert Lesson Title Here
Vocabulary
linear function

6. Course 3
12-5
Linear Functions
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.

7. Course 3
12-5
Linear Functions
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.

8. Additional Example 1 Continued
Course 3
12-5
Linear Functions
Additional Example 1 Continued
f(x) = mx + 2
4 = m(1) + 2 (x, y) = (1, 4)
4 = m + 2
– – 2
2 = m
The rule is f(x) = 2x + 2.

9. Linear Functions 12-5 Try This: Example 1
Course 3
12-5
Linear Functions
Try This: Example 1
Write the rule for the linear function. x y 2 -2 4 -4
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

10. Try This: Example 1 Continued
Course 3
12-5
Linear Functions
Try This: Example 1 Continued
f(x) = mx + 1
2 = m(5) + 1 (x, y) = (5, 2)
2 = 5m + 1
– – 1
1 = 5m 1 5
m =
The rule is f(x) = x + 1. 1 5

11. Course 3
12-5
Linear Functions
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. x y –2 5 –1 3 1
f(x) = mx + 1
–1 = m(1) + 1
–1 = m + 1
– –1
The rule is
f(x) = –2x + 1.
–2 = m

12. Course 3
12-5
Linear Functions
Additional Example 2B: Writing the Equation for a Linear Function from a Table
Write the rule for the linear function. B. x y –3 –8 –1 –2 1 4 3 10
Use two points, such as (1, 4) and (3, 10), to find the slope.
m = = = = 3
y2 – y1
x2 – x1
10 - 4
3 - 1 6 2
Substitute the x- and y-values of the point (1, 4) into f(x) = 3x + b, and solve for b.

13. Additional Example 2B Continued
Course 3
12-5
Linear Functions
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.

14. Linear Functions 12-5 Try This: Example 2A
Course 3
12-5
Linear Functions
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. x y –1 1 2 –2
f(x) = mx + 0
–1 = m(1) + 0
–1 = m
The rule is f(x) = –x.

15. Linear Functions 12-5 Try This: Example 2B
Course 3
12-5
Linear Functions
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. x y 5 1 6 2 7 –1 4
m = = = = 1
y2 – y1
x2 – x1
6 – 5
1 – 0 1
Substitute the x- and y-values of the point (0, 5) into f(x) = 1x + b, and solve for b.

16. Try This: Example 2 Continued
Course 3
12-5
Linear Functions
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.

17. Example 3: Money Application
Course 3
12-5
Linear Functions
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 After 12 video rentals, the cost will be f(12) = 1.5(12) + 15 = = $33.
– – 15
1.5 = m

18. Linear Functions 12-5 Try This: Example 3
Course 3
12-5
Linear Functions
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 After 10 book purchases, the cost will be
f(10) = 2(10) + 20 = = $40.
– – 20
2 = m

19. Insert Lesson Title Here
Course 3
12-5
Linear Functions
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. x –2 –1 1 2 y 8 5 –4
f(x) = –3x + 2 x –3 3 5 7 y
–10 –1 8 14 20
f(x) = 3x – 1
f(x) = 4.50x + 60; $172.50