Transforming Polynomial Functions

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Transforming Polynomial Functions
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
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Warm Up Let g be the indicated transformation of
f(x) 3x 1. Write the rule for g.
g(x) 3x 2
1. horizontal translation 1 unit right
2. vertical stretch by a factor of 2
g(x) 6x 2
g(x) 12x 1
3. horizontal compression by a factor of 4
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Objective
Transform polynomial functions.
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You can perform the same transformations on
polynomial functions that you performed on
quadratic and linear functions.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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) is a vertical compression of f(x).
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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.
g(x) is a horizontal stretch of f(x).
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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) f(x)
g(x) 4x4 6x2 1
g(x) is a vertical compression of f(x).
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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.
g(x) f( x)
g(x) x4 3x2 4
g(x) is a horizontal stretch of f(x).
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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.
g(x) 2(x 2)3 1
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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
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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.
g(x) 4(x 3)3 1
g(x) 4x3 36x2 108x 1
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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 andmove the
x-intercept 4 units left.
g(x) f(x 4)
g(x) 6(x 4)3 3
g(x) 8x3 96x2 384x 510
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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.
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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.
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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
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Lesson Quiz Part II
2.
Write a function that reflects f(x) 2x3 1
across the x-axis and shifts it 3 units down.
h(x) 2x3 4
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