13-4 13-4Linear LinearFunctions Functions Warm Up Problem of the Day Lesson Presentation Course Course 33 13-4 Linear Functions Warm Up Determine if each relationship represents a function. 1. yes 2. y = 3x2 – 1 yes 3. For the function f(x) = x2 + 2, find f(0), f(3), and f(–2). Course 3 2, 11, 6 13-4 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. Course 3 13-4 Linear Functions Learn to identify linear functions. Course 3 13-4 Linear Functions Vocabulary linear function function notation Course 3 13-4 Linear Functions A linear function 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. Reading Math f(x) is read “f of x.” f(1)is read “f of 1.” 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. Course 3 13-4 Linear Functions Additional Example 1: Identifying Linear Functions Determine whether the function f(x) = 2x3 is linear. f(x) = 2x3 y 4 Graph the function. 2 x -4 -2 2 -4 Course 3 4 f(x) = 2x3 does not represent a linear function because its graph is not in a straight line. 13-4 Linear Functions Check It Out: Example 1 Determine whether the function f(x) = -2x + 4 is linear. f(x) = –2x + 4 y 4 Graph the function. 2 x -4 -2 2 -2 -4 Course 3 4 f(x) = -2x + 4 does represent a linear function because its graph is in a straight line. It has a slope of -2 and a y-intercept of 4. 13-4 Linear Functions 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. Course 3 13-4 Linear Functions Additional Example 2A Continued f(x) = 4= 4= –2 2= mx + b m(1) + 2 m+2 –2 m The rule is f(x) = 2x + 2. Course 3 (x, y) = (1, 4) 13-4 Linear Functions Additional Example 2B: Writing the Equation for a Linear Function Write a rule for the linear function. x y Step 1 Locate two points. –3 –8 (1, 4) and (3, 10) –1 –2 Step 2 Find the slope m. 1 4 3 10 y2 – y1 10 – 4 6 m = x2 – x1 = = =3 3–1 2 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b. Course 3 13-4 Linear Functions Additional Example 2B Continued f(x) = mx + b 4 = 3(1) + b 4= 3+b –3 –3 1= b The rule is f(x) = 3x + 1. Course 3 (x, y) = (1, 4) 13-4 Linear Functions Check It Out: Example 2A Write a rule for the linear function. Step 1 Identify the y-intercept b from the graph. y 4 2 -4 -2 2 -4 Course 3 4 b=1 Step 2 Locate another point x 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. 13-4 Linear Functions Check It Out: Example 2A Continued f(x) = mx + b 2 = m(5) + 1 2 = 5m + 1 –1 –1 1 = 5m m=1 5 The rule is f(x) = 1 x + 1. 5 Course 3 (x, y) = (5, 2) 13-4 Linear Functions Check It Out: Example 2B Write a rule for the linear function. x y Step 1 Locate two points. 0 5 (0, 5) and (1, 6) 1 6 2 7 –1 4 Step 2 Find the slope m. y2 – y1 6 – 5 m = x2 – x1 = 1 – 0 = 1 1 =1 Step 3 Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b. Course 3 13-4 Linear Functions Check It Out: Example 2B Continued f(x) = mx + b 5 = 1(0) + b 5= b (x, y) = (0, 5) The rule is f(x) = x + 5. Course 3 13-4 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) = 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) = –15 – 15 1.5x + 15. After 12 video rentals, the 1.5 = m Course 3 cost will be f(12) = 1.5(12) + 15 = 18 + 15 = $33. 13-4 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) = mx + 20 The y-intercept is the cost to join, $20. With 1 book purchase the cost will be 22 = m(1) + 20 $22. 22 = m + 20 The rule for the function is –20 – 20 f(x) = 2x + 20. After 10 books 2=m purchases, the cost will be f(10) = 2(10) + 20 = 20 + 20 = $40. Course 3 13-4 Linear Functions Lesson Quiz: Part I Determine whether each function is linear. 1. f(x) = 4x2 not linear 2. f(x) = 3x + 1 linear Write the rule for the linear function. f(x) = 1 x - 1 2 Course 3 13-4 Linear Functions Lesson Quiz: Part II Write the rule for each linear function. 2. x y –3 0 3 –10 –1 8 5 7 14 20 f(x) = 3x – 1 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. f(x) = 4.50x + 60; $172.50 Course 3