Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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 function notation

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.

Additional Example 1: Identifying Linear Functions Determine whether the function f(x) = –2x3 is linear. If so, give the slope and y-intercept of the function’s graph. 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.

Check It Out: Example 1 Determine whether the function f(x) = –2x + 4 + x is linear. If so, give the slope and y-intercept of the function’s graph. 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.

Additional Example 2A: Writing the Equation for a Linear Function Write a rule for the 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.

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

Additional Example 2B: Writing the Equation for a Linear Function Write a rule for the linear function. Step 1 Locate two points. x y –3 –8 –1 –2 1 4 3 10 (1, 4) and (3, 10) Step 2 Find the slope m. m = = = = 3 y2 – y1 x2 – x1 10 – 4 3 – 1 6 2 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.

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

Write a rule for the linear function. Check It Out: Example 2A Write a rule for the linear function. Step 1 Identify the y-intercept b from the graph. x y 2 -2 4 -4 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.

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

Check It Out: Example 2B Write a rule for the linear function. Step 1 Locate two points. x y 5 1 6 2 7 –1 4 (0, 5) and (1, 6) Step 2 Find the slope m. m = = = = 1 y2 – y1 x2 – x1 6 – 5 1 – 0 1 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.

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.

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

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