Linear Functions 8-4 Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes
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Linear Functions8-4
Warm UpWarm Up
Lesson PresentationLesson Presentation
Problem of the DayProblem of the Day
Lesson QuizzesLesson Quizzes
Linear Functions8-4
Warm UpDetermine if each relationship represents a function.
1.
2. y = 3x2 – 1
3. For the function y = x2 + 2, find when
x = 0, x = 3, and x = –2.
yes
yes
2, 11, 6
Linear Functions8-4
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 Functions8-4
MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem.Also MA.8.A.1.1
Sunshine State Standards
Linear Functions8-4
Vocabulary
linear function
function notation
Linear Functions8-4
A linear function is a function that can be described by a linear equation. One way to write a linear function is by using function notation. If x represents the input value, then the and y represents the output value, the function notation for y is f(x), where f names the function.
Any linear function can be written in slope-intercept form f(x) = mx +b where m is the slope of the function’s graph and b is the y-intercept.
Linear Functions8-4
Determine whether the function f(x) = 2x3 is linear. If so, give the slope and y-intercept of the function’s graph.
Additional Example 1A: Identifying Linear Functions
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.
Linear Functions8-4
Determine whether the function f(x) = 3x + 3x + 3 is linear. If so, give the slope and y-intercept of the function’s graph.
Additional Example 1B: Identifying Linear Functions
f(x) = 3x +3x + 3
The function is linear because it can be written in the form f(x) = mx + b. The slope m is 6, and the y-intercept b is 3.
Write the equation in slope-intercept form.
f(x) = 6x + 3 Combine like terms.
Linear Functions8-4
Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.
Check It Out: Example 1A
f(x) = –2x + 4
m = –2; b = 4; f(x) = –2x + 4 is a linear function because it can be written in the form f(x) = mx + b.
Linear Functions8-4
Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.
Check It Out: Example 1B
f(x) =– + 41x
f(x) =– + 4 is not a linear function because
x appears in a denominator.
1x
Linear Functions8-4
Additional Example 2A: Writing the Equation for a 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.
Write a rule for the linear function.
Linear Functions8-4
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.
Linear Functions8-4
Additional Example 2B: Writing the Equation for a Linear Function
Step 1: Locate two points.
(1, 4) and (3, 10)
Step 2: Find the slope m.
Step 3: Substitute the x- and y-values of the point into the equation, f(x) = mx + b, and solve for b.
Write a rule for the linear function.
x y
–3 –8
–1 –2
1 4
3 10m = = = = 3
y2 – y1x2 – x1
10 – 43 – 1
62
Linear Functions8-4
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.
Linear Functions8-4
Check It Out: Example 2A
Write a rule for each linear function.
Linear Functions8-4
Check It Out: Example 2A Continued
b = 1; (5, 2): 2 = m(5) + 1
= m; f(x) = x + 115
15
Linear Functions8-4
Check It Out: Example 2B
b = –1; f(x) = 2x – 1
Write a rule for the linear function.
m = = 2 1 –(–1)1 – 0
x –2 –1 0 1 2
y –5 –3 –1 1 3
(0, –1) and (1, 1);
Linear Functions8-4
Example 3: Money ApplicationA video game club cost $15 to join. Each game 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) = 1.5x + 15
f(x) = 1.5(12) + 15
f(x) is the cost of renting games, and x is the number of games rented.
f(x) = 18 + 15
= 33
To write the rule, determine the slope and y-intercept.
m = 1.5b = 15
The rate of change is $1.50 per game.The cost to join is $15.
To rent 12 games as a member will cost $33.
Linear Functions8-4
Check It Out: Example 3A 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) = 2x + 20
f(10) = 2(10) + 20
= 20 + 20
= 40
rate of change = $2 per book; y-intercept is $20 membership fee;
The total cost of buying 10 books is $40.
Linear Functions8-4
Standard Lesson Quiz
Lesson Quizzes
Lesson Quiz for Student Response Systems
Linear Functions8-4
Determine whether each function is linear. If so, give the slope and y-intercept of the function’s graph.
1. f(x) = 4x2
2. f(x) = 3(x + 4)
Write the rule for the linear function.
3.
Lesson Quiz: Part I
linear; m = 3; b = 12
not linear
1 2
f(x) = x 1
Linear Functions8-4
Write the rule for each linear function.
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. Find a rule for the linear function that describes Andre's expenses for the day. Determine his expenses if he made 25 toys.
Lesson Quiz: Part II
f(x) = 3x – 1
f(x) = 4.50x + 60; $172.50
x –3 0 3 5 7
y –10 –1 8 14 20
Linear Functions8-4
1. Identify a function that is linear.
A. f(x) = 4x2
B. f(x) = 2(x2 + 1)
C. f(x) = 2(x + x)
D. f(x) = x2
Lesson Quiz for Student Response Systems
Linear Functions8-4
2. Identify a function that is not linear.
A. f(x) = x
B. f(x) = 0.5x
C. f(x) = 3(x + x) + 2
D. f(x) = 5x2
Lesson Quiz for Student Response Systems
Linear Functions8-4
3. Write the rule for the linear function.
A. f(x) = x + 3
B. f(x) = –x + 3
C. f(x) = x + 3
D. f(x) = 3x + 3
Lesson Quiz for Student Response Systems
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