Transcript chapter 8-4

8-4 Linear Functions
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
8-4 Linear Functions
Warm Up
Determine if each relationship represents
a function.
1.
yes
2. y = 3x2 – 1 yes
3. For the function y = x2 + 2, find when
x = 0, x = 3, and x = –2.
2, 11, 6
8-4 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.
8-4 Linear Functions
Sunshine State Standards
MA.8.A.1.2 Interpret the slope and the xand y-intercepts when graphing a linear
equation for a real-world problem.
Also MA.8.A.1.1
8-4 Linear Functions
Vocabulary
linear function
function notation
8-4 Linear Functions
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 slopeintercept form f(x) = mx +b where m is the slope
of the function’s graph and b is the y-intercept.
8-4 Linear Functions
Additional Example 1A: 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.
8-4 Linear Functions
Additional Example 1B: Identifying Linear Functions
Determine whether the function f(x) = 3x +
3x + 3 is linear. If so, give the slope and yintercept of the function’s graph.
f(x) = 3x +3x + 3
Write the equation in slopeintercept form.
f(x) = 6x + 3
Combine like terms.
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.
8-4 Linear Functions
Check It Out: Example 1A
Determine whether each function is linear. If so,
give the slope and y-intercept of the function’s
graph.
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.
8-4 Linear Functions
Check It Out: Example 1B
Determine whether each function is linear. If so,
give the slope and y-intercept of the function’s
graph.
f(x) =– 1 + 4
x
f(x) =– 1 + 4 is not a linear function because
x
x appears in a denominator.
8-4 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.
8-4 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.
(x, y) = (1, 4)
8-4 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.
8-4 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.
(x, y) = (1, 4)
8-4 Linear Functions
Check It Out: Example 2A
Write a rule for each linear function.
8-4 Linear Functions
Check It Out: Example 2A Continued
b = 1; (5, 2): 2 = m(5) + 1
1 = m; f(x) = 1 x + 1
5
5
8-4 Linear Functions
Check It Out: Example 2B
Write a rule for the linear function.
x
y
–2 –1 0
–5 –3 –1
1
1
2
3
(0, –1) and (1, 1); m = 1 –(–1) = 2
1–0
b = –1; f(x) = 2x – 1
8-4 Linear Functions
Example 3: Money Application
A 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.
To write the rule, determine the slope and y-intercept.
m = 1.5
b = 15
The rate of change is $1.50 per game.
The cost to join is $15.
f(x) = 1.5x + 15 f(x) is the cost of renting games, and
x is the number of games rented.
f(x) = 1.5(12) + 15
f(x) = 18 + 15
To rent 12 games as a member will
= 33
cost $33.
8-4 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.
rate of change = $2 per book; y-intercept is $20
membership fee;
f(x) = 2x + 20
f(10) = 2(10) + 20
= 20 + 20
= 40
The total cost of buying 10 books is $40.
8-4 Linear Functions
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
8-4 Linear Functions
Lesson Quiz: Part I
Determine whether each function is linear.
If so, give the slope and y-intercept of the
function’s graph.
1. f(x) = 4x2
not linear
2. f(x) = 3(x + 4) linear; m = 3; b = 12
Write the rule for the linear function.
3.
f(x) = 1 x - 1
2
8-4 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 made 25
toys.
f(x) = 4.50x + 60; $172.50
8-4 Linear Functions
Lesson Quiz for Student Response Systems
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
8-4 Linear Functions
Lesson Quiz for Student Response Systems
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
8-4 Linear Functions
Lesson Quiz for Student Response Systems
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