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6-8
TransformingPolynomial
PolynomialFunctions
Functions
6-8 Transforming
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
6-8 Transforming Polynomial Functions
Warm Up
Let g be the indicated transformation of
f(x) = 3x + 1. Write the rule for g.
1. horizontal translation 1 unit right
g(x) = 3x – 2
2. vertical stretch by a factor of 2
g(x) = 6x + 2
3. horizontal compression by a
factor of 4
g(x) = 12x + 1
Holt Algebra 2
6-8 Transforming Polynomial Functions
Objective
Transform polynomial functions.
Holt Algebra 2
6-8 Transforming Polynomial Functions
You can perform the same transformations on
polynomial functions that you performed on
quadratic and linear functions.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 1A: Translating a Polynomial Function
For f(x) = x3 – 6, write the rule for each
function and sketch its graph.
g(x) = f(x) – 2
g(x) = (x3 – 6) – 2
g(x) = x3 – 8
To graph g(x) = f(x) – 2,
translate the graph of f(x)
2 units down.
This is a vertical
translation.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 1B: Translating a Polynomial Function
For f(x) = x3 – 6, write the rule for each
function and sketch its graph.
h(x) = f(x + 3)
h(x) = (x + 3)3 – 6
To graph h(x) = f(x + 3),
translate the graph 3 units
to the left.
This is a horizontal
translation.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 1a
For f(x) = x3 + 4, write the rule for each
function and sketch its graph.
g(x) = f(x) – 5
g(x) = (x3 + 4) – 5
g(x) = x3 – 1
To graph g(x) = f(x) – 5,
translate the graph of f(x)
5 units down.
This is a vertical
translation.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 1b
For f(x) = x3 + 4, write the rule for each
function and sketch its graph.
g(x) = f(x + 2)
g(x) = (x + 2)3 + 4
g(x) = x3 + 6x2 + 12x + 12
To graph g(x) = f(x + 2),
translate the graph 2 units
left.
This is a horizontal
translation.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 2A: Reflecting Polynomial Functions
Let f(x) = x3 + 5x2 – 8x + 1. Write a function
g that performs each transformation.
Reflect f(x) across the x-axis.
g(x) = –f(x)
g(x) = –(x3 + 5x2 – 8x + 1)
g(x) = –x3 – 5x2 + 8x – 1
Check Graph both
functions. The graph
appears to be a reflection.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 2B: Reflecting Polynomial Functions
Let f(x) = x3 + 5x2 – 8x + 1. Write a function
g that performs each transformation.
Reflect f(x) across the y-axis.
g(x) = f(–x)
g(x) = (–x)3 + 5(–x)2 – 8(–x) + 1
g(x) = –x3 + 5x2 + 8x + 1
Check Graph both
functions. The graph
appears to be a reflection.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 2a
Let f(x) = x3 – 2x2 – x + 2. Write a function g
that performs each transformation.
Reflect f(x) across the x-axis.
g(x) = –f(x)
g(x) = –(x3 – 2x2 – x + 2)
g(x) = –x3 + 2x2 + x – 2
Check Graph both
functions. The graph
appears to be a reflection.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 2b
Let f(x) = x3 – 2x2 – x + 2. Write a function g
that performs each transformation.
Reflect f(x) across the y-axis.
g(x) = f(–x)
g(x) = (–x)3 – 2(–x)2 – (–x) + 2
g(x) = –x3 – 2x2 + x + 2
Check Graph both
functions. The graph
appears to be a reflection.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 3A: Compressing and Stretching Polynomial
Functions
Let f(x) = 2x4 – 6x2 + 1. Graph f and g on the
same coordinate plane. Describe g as a
transformation of f.
g(x) = 1 f(x)
2
g(x) = 1 (2x4 – 6x2 + 1)
2
g(x) = x4 – 3x2 + 1
2
g(x) is a vertical
compression of f(x).
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 3B: Compressing and Stretching Polynomial
Functions
Let f(x) = 2x4 – 6x2 + 1. Graph f and g on the
same coordinate plane. Describe g as a
transformation of f.
h(x) = f( 1
x)
3
h(x) = 2( 1 x)4 – 6( 1 x)2 + 1
3
3
2 4
2 2
h(x) = 81
x – 3
x +1
g(x) is a horizontal stretch
of f(x).
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 3a
Let f(x) = 16x4 – 24x2 + 4. Graph f and g on the
same coordinate plane. Describe g as a
transformation of f.
g(x) = 1
f(x)
4
g(x) = 1 (16x4 – 24x2 + 4)
4
g(x) = 4x4 – 6x2 + 1
g(x) is a vertical
compression of f(x).
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 3b
Let f(x) = 16x4 – 24x2 + 4. Graph f and g on the
same coordinate plane. Describe g as a
transformation of f.
h(x) = f( 1
x)
2
h(x) = 16( 1 x)4 – 24( 1 x)2 + 4
2
2
h(x) = x4 – 3x2 + 4
g(x) is a horizontal stretch
of f(x).
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 4A: Combining Transformations
Write a function that transforms f(x) = 6x3 – 3
in each of the following ways. Support your
solution by using a graphing calculator.
Compress vertically by a factor
of 1 , and shift 2 units right.
3
g(x) = 1 f(x – 2)
3
g(x) = 1 (6(x – 2)3 – 3)
3
g(x) = 2(x – 2)3 – 1
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 4B: Combining Transformations
Write a function that transforms f(x) = 6x3 – 3
in each of the following ways. Support your
solution by using a graphing calculator.
Reflect across the y-axis and
shift 2 units down.
g(x) = f(–x) – 2
g(x) = (6(–x)3 – 3) – 2
g(x) = –6x3 – 5
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 4a
Write a function that transforms f(x) = 8x3 – 2
in each of the following ways. Support your
solution by using a graphing calculator.
Compress vertically by a
factor of 1 , and move the
2
x-intercept 3 units right.
g(x) = 1 f(x – 3)
2
g(x) = 1 (8(x – 3)3 – 2
2
g(x) = 4(x – 3)3 – 1
g(x) = 4x3 – 36x2 + 108x – 1
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 4b
Write a function that transforms f(x) = 6x3 – 3
in each of the following ways. Support your
solution by using a graphing calculator.
Reflect across the x-axis and
move the x-intercept 4 units
left.
g(x) = –f(x + 4)
g(x) = –6(x + 4)3 – 3
g(x) = –8x3 – 96x2 – 384x – 510
Holt Algebra 2
6-8 Transforming Polynomial Functions
Example 5: Consumer Application
The number of skateboards sold per month can
be modeled by f(x) = 0.1x3 + 0.2x2 + 0.3x +
130, where x represents the number of months
since May. Let g(x) = f(x) + 20. Find the rule for
g and explain the meaning of the transformation
in terms of monthly skateboard sales.
Step 1 Write the new rule.
The new rule is g(x) = f(x) + 20
g(x) = 0.1x3 + 0.2x2 + 0.3x + 130 + 20
g(x) = 0.1x3 + 0.2x2 + 0.3x + 150
Step 2 Interpret the transformation.
The transformation represents a vertical shift 20 units
up, which corresponds to an increase in sales of 20
skateboards per month.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Check It Out! Example 5
The number of bicycles sold per month can be
modeled by f(x) = 0.01x3 + 0.7x2 + 0.4x + 120,
where x represents the number of months since
January. Let g(x) = f(x – 5). Find the rule for g
and explain the meaning of the transformation in
terms of monthly skateboard sales.
Step 1 Write the new rule.
The new rule is g(x) = f(x – 5).
g(x) = 0.01(x – 5)3 + 0.7(x – 5)2 + 0.4(x – 5) + 120
g(x) = 0.01x3 + 0.55x2 – 5.85x + 134.25
Step 2 Interpret the transformation.
The transformation represents the number of sales
since March.
Holt Algebra 2
6-8 Transforming Polynomial Functions
Lesson Quiz: Part I
1. For f(x) = x3 + 5, write the rule for
g(x) = f(x – 1) – 2 and sketch its graph.
g(x) = (x – 1)3 + 3
2. Write a function that
reflects f(x) = 2x3 + 1 across
the x-axis and shifts it 3 units
down.
h(x) = –2x3 – 4
Holt Algebra 2
6-8 Transforming Polynomial Functions
Lesson Quiz: Part II
2.
3. The number of videos sold per month can be
modeled by f(x) = 0.02x3 + 0.6x2 + 0.2x +
125, where x represents the number of
months since July. Let g(x) = f(x) – 15. Find
the rule for g and explain the meaning of the
transformation in terms of monthly video
sales. 0.02x3 + 0.6x2 + 0.2x + 110; vertical
shift 15 units down; decrease of 15
units per month
Holt Algebra 2
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