1. Linear Functions 8-4 Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. Linear Functions 8-4 Warm Up Determine if each relationship represents a function. 1. 2. y = 3x 2 – 1 3. For the function y = x 2 + 2, find when x = 0, x = 3, and x = –2. yes 2, 11, 6

3. Linear Functions 8-4 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.

4. Linear Functions 8-4 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

5. Linear Functions 8-4 Vocabulary linear function function notation

6. Linear Functions 8-4 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.

7. Linear Functions 8-4 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 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.

8. Linear Functions 8-4 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 + 3Combine like terms.

9. Linear Functions 8-4 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.

10. Linear Functions 8-4 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) =– + 4 1x1x f(x) =– + 4 is not a linear function because x appears in a denominator. 1x1x

11. Linear Functions 8-4 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.

12. Linear Functions 8-4 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.

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

14. Linear Functions 8-4 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.

15. Linear Functions 8-4 Check It Out: Example 2A Write a rule for each linear function.

16. Linear Functions 8-4 Check It Out: Example 2A Continued b = 1; (5, 2): 2 = m(5) + 1 = m; f(x) = x + 1 1 5 1 5

17. Linear Functions 8-4 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–2–1012 y–5–3–113 (0, –1) and (1, 1);

18. Linear Functions 8-4 Example 3: Money Application A 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 = 1.5 b = 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.

19. Linear Functions 8-4 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(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.

20. Linear Functions 8-4 Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

21. Linear Functions 8-4 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

22. Linear Functions 8-4 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 –30357 y –10–181420

23. Linear Functions 8-4 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

24. Linear Functions 8-4 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) = 5x 2 Lesson Quiz for Student Response Systems

25. Linear Functions 8-4 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