1. 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 Warm Up

2. Pre-Algebra Linear Functions 12-5

3. Learn to identify linear functions.

4. linear function Vocabulary

5. 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.

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

7. 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. Example Continued

8. Write the rule for the linear function. x y 2 2-2 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 + 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 Try This

9. 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 Try This Continued

10. 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 xy –25 –13 01 1 –2 = m The rule is f(x) = –2x + 1. –1 = m + 1 –1 Example: Writing the Equation for a Linear Function from a Table

11. 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. xy –3–8 –1–2 14 310 m = = = = 3 y 2 – y 1 x 2 – x 1 10 - 4 3 - 1 6 2 Example: Writing the Equation for a Linear Function from a Table

12. f(x) = 3x + b 4 = 3(1) + b (x, y) = (1, 4) 4 = 3 + b –3 1 = b The rule is f(x) = 3x + 1. Example Continued

13. 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 xy 00 –11 1 2–2 –1 = m The rule is f(x) = –x. Try This

14. 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. xy 05 16 27 –14 m = = = = 1 y 2 – y 1 x 2 – x 1 6 – 5 1 – 0 1 1 Try This

15. f(x) = mx + b 5 = 1(0) + b (x, y) = (0, 5) 5 = b The rule is f(x) = x + 5. Try This Continued

16. 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

17. 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) + 20 The 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

18. 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 –2–1012 y 852 –4 x–30357 y –10–181420 Lesson Quiz