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Learn to identify linear functions, write equations from graphs or tables, and apply them to real-world scenarios. Practice determining if relationships are functions and solving problems. Explore the basics of slopes and y-intercepts in linear functions.
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Linear Functions 12-5 Course 3 Warm Up Problem of the Day Lesson Presentation
Linear Functions 12-5 Course 3 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
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 Course 3 Learn to identify linear functions.
Linear Functions 12-5 Course 3 Insert Lesson Title Here Vocabulary linear function
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 y 4 2 x -4 -2 2 4 -4 Course 3 Try This: Example 1 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 = 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
Linear Functions 12-5 1 1 5 5 m = The rule is f(x) = x + 1. Course 3 Try This: Example 1 Continued f(x) = mx + 1 2 = m(5) + 1 (x, y) = (5, 2) 2 = 5m + 1 – 1 – 1 1 = 5m
Linear Functions 12-5 Course 3 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. f(x) = mx + 1 –1 = m(1) + 1 –1 = m + 1 –1 –1 The rule is f(x) = –2x + 1. –2 = m
Linear Functions 12-5 y2 – y1 m = = = = 3 x2 – x1 10 - 4 3 - 1 6 2 Course 3 Additional Example 2B: Writing the Equation for a Linear Function from a Table 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.
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 Course 3 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. f(x) = mx + 0 –1 = m(1) + 0 –1 = m The rule is f(x) = –x.
Linear Functions 12-5 y2 – y1 m = = = = 1 x2 – x1 6 – 5 1 – 0 1 1 Course 3 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. Substitute the x- and y-values of the point (0, 5) into f(x) = 1x + b, and solve for b.
Linear Functions 12-5 Course 3 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.
Linear Functions 12-5 Course 3 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
Linear Functions 12-5 Course 3 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
Linear Functions 12-5 Course 3 Insert Lesson Title Here 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. f(x) = –3x + 2 f(x) = 3x – 1 f(x) = 4.50x + 60; $172.50
