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Pre-Algebra
12-5 Linear Functions12-5 Linear Functions
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
12-5 Linear Functions
Warm UpDetermine 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
Pre-Algebra
12-5 Linear Functions
Pre-Algebra
12-5 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.
Pre-Algebra
12-5 Linear Functions
Learn to identify linear functions.
Pre-Algebra
12-5 Linear Functions
Vocabulary
linear function
Pre-Algebra
12-5 Linear Functions
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.
Pre-Algebra
12-5 Linear Functions
Write the rule for the linear function.
Additional Example 1: Writing the Equation for a Linear Function from a Graph
Use the equation f(x) = mx + b. To find b, identify the y-intercept from the graph.
b = 2
f(x) = mx + 2Locate 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.
Pre-Algebra
12-5 Linear Functions
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.
Pre-Algebra
12-5 Linear Functions
Write the rule for the linear function.
Try This: Example 1
x
y
2
2-2
4
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 + 1Locate 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
Pre-Algebra
12-5 Linear Functions
Try This: Example 1 Continued
f(x) = mx + 1
2 = m(5) + 1 (x, y) = (5, 2)
2 = 5m + 1– 1 – 1
1 = 5m15m =
The rule is f(x) = x + 1.15
Pre-Algebra
12-5 Linear Functions
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
x y
–2 5
–1 3
0 1
1 –1
–2 = m
The rule isf(x) = –2x + 1.
–1 = m + 1–1 –1
Pre-Algebra
12-5 Linear Functions
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.
x y
–3 –8
–1 –2
1 4
3 10
m = = = = 3 y2 – y1x2 – x1
10 - 43 - 1
62
Pre-Algebra
12-5 Linear Functions
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.
Pre-Algebra
12-5 Linear Functions
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
x y
0 0
–1 1
1 –1
2 –2
–1 = m
The rule is f(x) = –x.
Pre-Algebra
12-5 Linear Functions
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.
x y
0 5
1 6
2 7
–1 4
m = = = = 1 y2 – y1x2 – x1
6 – 51 – 0
11
Pre-Algebra
12-5 Linear Functions
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.
Pre-Algebra
12-5 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
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.
Pre-Algebra
12-5 Linear Functions
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
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.
Pre-Algebra
12-5 Linear Functions
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.
Lesson Quiz
f(x) = 3x – 1
f(x) = –3x + 2
f(x) = 4.50x + 60; $172.50
x –2 –1 0 1 2
y 8 5 2 –1 –4
x –3 0 3 5 7
y –10 –1 8 14 20