Linear Functions 12-5 Warm Up Problem of the Day Lesson Presentation Pre-Algebra

Linear Functions 12-5 Warm Up Pre-Algebra 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

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.

Learn to identify linear functions.

Vocabulary linear function

The graph of a linear function is a line 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.

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.

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.

Write the rule for the linear function. 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

Try This: Example 1 Continued f(x) = mx + 1 2 = m(5) + 1 (x, y) = (5, 2) 2 = 5m + 1 – 1 – 1 1 = 5m 1 5 m = The rule is f(x) = x + 1. 1 5

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 –1 The rule is f(x) = –2x + 1. –2 = m

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.

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.

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.

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.

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.

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

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

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