1. Linear Functions Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. Linear Functions Warm Up Determine if each relationship represents a function. 1. 2. y = 3x 2 – 1 3. For the function f(x) = x 2 + 2, find f(0), f(3), and f(–2). yes 2, 11, 6

3. 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. Linear Functions Learn to identify linear functions.

5. Linear Functions Vocabulary linear function function notation

6. Linear Functions A linear function is a function that can be described by a linear equation. You can use function notation to show that the output value of the function f, written f(x), corresponds to the input value x. Any linear function can be written in slope- intercept form f(x) = mx +b.

7. Linear Functions Determine whether the function f(x) = –2x 3 is linear. If so, give the slope and y-intercept of the function’s graph. Additional Example 1: 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.

8. Linear Functions Determine whether the function f(x) = –2x + 4 + x is linear. If so, give the slope and y-intercept of the function’s graph. Check It Out: Example 1 f(x) = –2x + 4 + x f(x) = –x + 4 The function is linear because it can be written in the form f(x) = mx + b. The slope is –1 and the y-intercept is 4.

9. Linear Functions 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.

10. Linear Functions Additional Example 2A Continued f(x) = mx + b 4 = m(1) + 2(x, y) = (1, 4) 4 = m + 2 – 2 2 = m The rule is f(x) = 2x + 2.

11. Linear Functions 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. xy –3–8 –1–2 14 310 m = = = = 3 y 2 – y 1 x 2 – x 1 10 – 4 3 – 1 6 2

12. Linear Functions Additional Example 2B Continued f(x) = mx + b 4 = 3(1) + b(x, y) = (1, 4) 4 = 3 + b – 3 1 = b The rule is f(x) = 3x + 1.

13. Linear Functions Check It Out: Example 2A Step 1 Identify the y-intercept b from the graph. b = 1 Step 2 Locate another point on the graph, such as (5, 2). 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. x y 2 2-2 4 4-4

14. Linear Functions Check It Out: Example 2A Continued f(x) = mx + b 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

15. Linear Functions Check It Out: Example 2B Step 1 Locate two points. (0, 5) and (1, 6) 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. xy 05 16 27 –14 m = = = = 1 y 2 – y 1 x 2 – x 1 6 – 5 1 – 0 1 1

16. Linear Functions Check It Out: Example 2B Continued f(x) = mx + b 5 = 1(0) + b (x, y) = (0, 5) 5 = b The rule is f(x) = x + 5.

17. 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) = 1.5x + 15 f(x) = 1.5(12) + 15 f(x) is the cost of renting movies, and x is the number of videos rented. f(x) = 18 + 15 = 33 To write the rule, determine the slope and y-intercept. m = 1.5 b = 15 The rate of change is $1.50 per video. The cost to join is $15. To rent 12 videos as a member will cost $33.

18. Linear Functions Check It Out: 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) = 2x + 20 f(x) = 2(10) + 20 f(x) is the cost of buying books, and x is the number of books purchased. f(x) = 20 + 20 = 40 To write the rule, determine the slope and y-intercept. m = 2 b = 20 The rate of change is $2 per book. The cost to join is $20. To buy 10 books as a member will cost $40.

19. Linear Functions Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

20. Linear Functions Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph. 1. f(x) = 4x 2 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 1212 f(x) = x  1

21. Linear Functions 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 sold 25 toys. Lesson Quiz: Part II f(x) = 3x – 1 f(x) = 4.50x + 60; $172.50 x –30357 y –10–181420

22. Linear Functions 1. Identify a function that is linear. A. f(x) = 4x 2 B. f(x) = 2(x 2 + 1) C. f(x) = 2(x + x) D. f(x) = x 2 Lesson Quiz for Student Response Systems

23. Linear Functions 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) = 5x 2 Lesson Quiz for Student Response Systems

24. Linear Functions 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