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Linear Functions
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
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).
2, 11, 6
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.
Linear Functions
Learn to identify linear functions.
Linear Functions
Vocabulary
linear function
function notation
Linear Functions
A linear function is a function that 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.
Any linear function can be written in slopeintercept form f(x) = mx +b.
Linear Functions
Additional Example 1: 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.
Linear Functions
Check It Out: Example 1
Determine whether the function f(x) = –2x + 4 + x
is linear. If so, give the slope and y-intercept of the
function’s graph.
f(x) = –2x + 4 + x
f(x) = –x + 4
The function is linear because it can be
written in the form f(x) = mx + b. The slope
is –1 and the y-intercept is 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.
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)
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.
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)
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
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.
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
(x, y) = (5, 2)
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.
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.
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.
To write the rule, determine the slope and y-intercept.
m = 1.5
b = 15
The rate of change is $1.50 per video.
The cost to join is $15.
f(x) = 1.5x + 15 f(x) is the cost of renting movies, and
x is the number of videos rented.
f(x) = 1.5(12) + 15
f(x) = 18 + 15
To rent 12 videos as a member will
= 33
cost $33.
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.
To write the rule, determine the slope and y-intercept.
m=2
b = 20
f(x) = 2x + 20
The rate of change is $2 per book.
The cost to join is $20.
f(x) is the cost of buying books, and
x is the number of books purchased.
f(x) = 2(10) + 20
f(x) = 20 + 20
To buy 10 books as a member will
= 40
cost $40.