9-3
Functions
9-3 Transforming
Transforming
Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
9-3 Transforming Functions
Warm Up
A rental car costs $45 per day plus
$0.10 for every mile over 200.
1. Find the cost of renting the car for a day and
driving 250 miles. $50
2. Write a function of d, the number of miles
driven in a day, to describe the cost of renting
the car for one day?
45
if 0 ≤ d ≤ 200
C(d) =
45 + 0.1(d – 200) if d > 200
Holt Algebra 2
9-3 Transforming Functions
Objectives
Transform functions.
Recognize transformations of functions.
Holt Algebra 2
9-3 Transforming Functions
In previous lessons, you learned how to
transform several types of functions. You can
transform piecewise functions by applying
transformations to each piece independently.
Recall the rules for transforming functions given
in the table.
Holt Algebra 2
9-3 Transforming Functions
Holt Algebra 2
9-3 Transforming Functions
Caution
Horizontal translations change both the rules and
the intervals of piecewise functions. Vertical
translations change only the rules.
Holt Algebra 2
9-3 Transforming Functions
Example 1: Transforming Piecewise Functions
–
Given f(x) =
1
x
2
1
x2
2
if x < 0
if x ≥ 0
write the
rule g(x), a vertical stretch by a factor of 3.
Holt Algebra 2
9-3 Transforming Functions
Example 1 Continued
Each piece of f(x) must be vertically stretched by
a factor of 3. Replace every y in the function by
3y, and simplify.
g(x) = 3f(x) =
1
3(–
x)
2
1
3(
x2)
2
–
=
Holt Algebra 2
3
2
x
3
x2
2
if x < 0
if x ≥ 0
if x < 0
if x ≥ 0
9-3 Transforming Functions
Example 1 Continued
Check Graph both functions to support your answer.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 1
x2
if x ≤ 0
Given f(x) =
write the rule
x – 3 if x > 0
for g(x), a horizontal stretch of f(x) by a
factor of 2.
Each piece of f(x) must be stretched by a factor
of 2 units.
1
2
x
if x ≤ 0
1
2
g(x) = f(
x) =
1
b
x – 3 if x > 0
=
Holt Algebra 2
2
x2
2
x
–3
2
if x ≤ 0
if x > 0
9-3 Transforming Functions
Check It Out! Example 1 Continued
Check Graph both functions to support your answer.
Holt Algebra 2
9-3 Transforming Functions
When functions are transformed, the intercepts
may or may not change. By identifying the
transformations, you can determine the intercepts,
which can help you graph a transformed function.
Holt Algebra 2
9-3 Transforming Functions
Holt Algebra 2
9-3 Transforming Functions
Holt Algebra 2
9-3 Transforming Functions
Example 2A: Identifying Intercepts
Identify the x- and y-intercepts of f(x).
Without graphing g(x), identify its x- and yintercepts.
f(x) =–2x – 4 ; g(x) =
Find the intercepts of the original function.
f(0) = –2(0) – 4 = – 4
0 = –2x – 4
f(0) = –4
–2 = x
Holt Algebra 2
9-3 Transforming Functions
Example 2A Continued
The y-intercept is –4, and the x-intercept is –2.
Note that g(x) is a horizontal stretch of f(x) by a
factor of 2. So the y-intercept of g(x) is also –4.
The x-intercept is 2(–2), or –4.
Check A graph
supports
your answer.
Holt Algebra 2
9-3 Transforming Functions
Example 2B: Identify Intercepts
f(x) = x2 – 1; g(x) = f(–x)
From the graph of f(x),
the y-intercept is –1,
and the x-intercepts
are –1 and 1.
Note that g(x) is a horizontal Check A graph supports
your answer.
reflection across the y-axis. So
the x-intercepts of g(x) will be
–1(–1) and –1(1), or 1 and –1.
The y-intercept is unchanged
at –1.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 2a
Identify the x- and y-intercepts of f(x).
Without graphing g(x), identify its x- and yintercepts.
f(x) =
2
x + 4 and g(x) = –f(x)
3
Find the intercepts of the original functions.
f(0) = 2 (0) + 4
3
f(0) = 4
Holt Algebra 2
0=
2
x+4
3
–6 = x
9-3 Transforming Functions
Check It Out! Example 2a Continued
The y-intercept is 4, and the x-intercept is –6.
Note that g(x) is a reflection of f(x) across the
x-axis. So the x-intercept of g(x) is also –6. The
y-intercept is –1(4), or –4.
Check A graph supports
your answer.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 2b
f(x) = x2 – 9 and g(x) =
f(x)
From the graph of f(x),
the y-intercept is –9,
and the x-intercepts
are ±3.
Note that g(x) is a vertical
1
Check A graph supports
compression by a factor of
.
3
So the x-intercept of g(x) will
be ±3. The y-intercept of
1
g(x) will be
(–9) = –3.
3
Holt Algebra 2
your answer
9-3 Transforming Functions
Remember!
The factor for horizontal stretches and
compressions is the reciprocal of the coefficient
in the equation.
1
3
2
Holt Algebra 2
=
2
3
9-3 Transforming Functions
Example 3: Combining Transformations
1
Given f(x) =
(x – 2)2 and g(x) = 2f(x) – 3 and
3
graph g(x).
Step 1 Graph f(x). The graph of f(x) has y-intercept
(0, 4 ) and x-intercept (2, 0).
3
Holt Algebra 2
9-3 Transforming Functions
Example 3 Continued
Step 2 Analyze each transformation one at a time.
The first transformation is a vertical stretch by a
factor of 2. After the vertical stretch, the x-intercept
will remain 2, but the y-intercept will be 8 .
3
The second transformation is a vertical translation of
3 units down. Use a table to shift each identified
point down 3 units.
Holt Algebra 2
9-3 Transforming Functions
Example 3 Continued
Intercept
Points
(2, 0)
Shifted
(2, –3)
8 )
3
(0, – 1 )
3
(0,
Step 3 Graph the final result.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 3
1
Given f(x) = 2x – 4 and g(x)= –
f(x), graph
2
g(x).
Step 1 Graph f(x). The graph of f(x) has y-intercept
(0, –3) and x-intercept (2, 0).
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 3 Continued
Step 2 Analyze each transformation one at a time.
The first transformation is a vertical compression
by 1 . After the vertical compression, the x2
intercept is (2, 0) but the y-intercept is (0, –3).
The second transformation is a reflection across the
x-axis. The x-intercept remains (2, 0) but the yintercept will be (0, 1.5).
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 3 Continued
Step 3
Holt Algebra 2
Graph the final result.
9-3 Transforming Functions
Example 4: Problem-Solving Application
Coco’s Coffee charges different prices
based on the number of pounds
purchased. The pricing scale is modeled
by the function below, where w is the
weight in pounds purchased.
p(w) =
Holt Algebra 2
9w
if 0 < w < 3
27 + 7.5(w–3)
if 3 ≤ w < 6
49.5 + 6(w–6)
if w ≥ 6
9-3 Transforming Functions
Example 4 Continued
Orders placed directly through the Web site
are discounted by 1 , but a shipping fee of
3
$2.50 is added. Write a pricing function for
orders placed through the Web site.
Holt Algebra 2
9-3 Transforming Functions
Example 4 Continued
1
Understand the Problem
The new price function will include two changes, a
discount by 1 and an additional shipping fee of
3
$2.50. The discount is equivalent to multiplying all
of the parts of the function by 2 . This will be a
3
2
vertical compression by a factor of
. The
3
shipping price will be a vertical translation of 2.5
units up.
Holt Algebra 2
9-3 Transforming Functions
Example 4 Continued
2
Make a Plan
Perform each transformation, one at a time,
and then write the new rule.
Holt Algebra 2
9-3 Transforming Functions
Example 4 Continued
3
Solve
First find the prices after the discount for Web
purchases.
pdiscount(w) =
2
3
p(w) =
2
3
(9w)
2
3
(27 + 7.5(w – 3)) if 3 ≤ w < 6
2
3
Holt Algebra 2
if 0 < w < 3
(49.5 + 6(w – 6)) if w ≥ 6
Multiply
all parts
of the
function
by 2 .
3
9-3 Transforming Functions
Example 4 Continued
3
Solve Continued
=
Holt Algebra 2
6w
if 0 < w < 3
18 + 5(w – 3)
if 3 ≤ w < 6
33 + 4(w – 6)
if w ≥ 6
9-3 Transforming Functions
Example 4 Continued
3
Solve Continued
Then find the prices after applying the shipping fees.
Pw(w) = pdiscount(w) + 2.5 =
=
Holt Algebra 2
6w + 2.5
if 0 < w < 3
18 + 5(w – 3) + 2.5
if 3 ≤ w < 6
33 + 4(w – 6) + 2.5
if w ≥ 6
6w + 2.5
if 0 < w < 3
20.5 + 5(w – 3)
if 3 ≤ w < 6
35.5 + 4(w – 6)
if w ≥ 6
9-3 Transforming Functions
Example 4 Continued
4
Look Back
Check your answer by trying a few values. For 9
pounds of coffee, the original fee would have been
1
$67.50. A
discount plus $2.50 would amount to
3
$47.50. Evaluate the function for x = 9 to check.
Pw(9) = {35.5 + 4(9 – 6) = 47.50 
Continue by checking each piece of the function.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 4
A movie theater charges $5 for children
under 12 and $7.50 for anyone 12 and over.
The theater decides to increase its prices by
20%. It charges an additional $0.50 fee for
online ticket purchases. Write an equation
for the online ticket prices.
f(x) =
Holt Algebra 2
5x
if 0 < x < 12
7.5x
if x ≥ 12
9-3 Transforming Functions
Check It Out! Example 4 Continued
1
Understand the Problem
Two changes, a 20% increase plus an additional fee
of $0.50 for online prices. The increase is equivalent
to multiplying all parts by 120% or 1.2. This is a
vertical stretch by a factor of 1.2. The online fee will
be a vertical translation of 0.50 units up.
2
Make a Plan
Perform each transformation, one at a time, and
then write the new rule.
Holt Algebra 2
9-3 Transforming Functions
Check It Out! Example 4 Continued
3
Solve
First find the increase.
(1.2)5x
fincrease(x) = (1.2)f(x) =
if 0 < x < 12
(1.2)7.5x if x ≥ 12
Then find the online fees.
fonline(x) = f(x + 0.50) =
Holt Algebra 2
6.50x
if 0 < x < 12
9.50x
if x ≥ 12
Multiply all parts
of the function
by 1.2.
9-3 Transforming Functions
Check It Out! Example 4 Continued
4
Look Back
Check your answer by trying a few values. For 1
child age 10 and 1 adult age 35, the original fee
would have been $12.50. A 20% increase plus
$0.50 per ticket will be $16. Check the function x =
1 to check.
fonline(1) + fonline(1) = 6.50(1) + 9.50(1) = 16 
Continue by checking each piece of the function.
Holt Algebra 2
9-3 Transforming Functions
Lesson Quiz: Part I
Consider the functions
1
f(x) =
x2 – 2, g(x) = 4f(x), and
2
h(x) = f( 1 x) + 3 .
3
1. Identify the intercepts of f(x) and g(x).
f(x): x-ints. = –2 and 2, y-int. = –2
g(x): x-ints. = –2 and 2, y-int. = –8
2. Graph f(x) and h(x).
Holt Algebra 2
9-3 Transforming Functions
Lesson Quiz: Part II
3. Ticket prices to a theater are modeled by the
function below, where a is a person’s age in
years.
4.50
if 0 ≤ a < 12
p(a) =
7.00
if a ≥ 12
They plan to raise all prices by $1, but then they
are going to offer a 25% discount to persons 55
and older. Write a function for their new prices.
p(a) =
Holt Algebra 2
5.50
if 0 ≤ a < 12
8.00
if 12 ≤ a < 55
6.00
if a ≥ 55