Name ________________________________________ Date __________________ Class__________________
LESSON
3-8
Reteach
Transforming Polynomial Functions
Translations of polynomial functions shift the graph of the function right, left, up, or down.
Vertical Translation
If f (x) is a polynomial function,
g(x)  f (x)  k is a vertical
translation of f (x).
Example: f (x)  x 3  2
Think: Add to y, go high.
f (x) shifts up for k  0.
f (x) shifts down for k  0.
To graph g(x), move
the graph of f (x)
5 units down.
Vertical translation 5 units down
g (x )  f (x )  5
g (x )  x 3  2  5
g (x )  x 3  3
Horizontal Translation
If f(x) is a polynomial function,
g(x)  f (x  h) is a horizontal
translation of f (x).
Example: f (x)  x 3  2
Think: Add to x, go west.
f (x) shifts right for h  0.
f (x) shifts left for h  0.
Horizontal translation 4 units left

g(x)  f x  ( 4)
To graph g(x), move
the graph of f (x)
4 units left.

g(x)  (x  4) 3  2
For f (x)  x 3  2, write the rule for each function and sketch its graph.
1. g(x)  f (x)  1
2. g(x)  f (x  3)
Translate f (x) 1 unit ______________.
Translate f (x) 3 units ______________.
g(x)  _____________________
g(x)  _____________________
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3-62
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
3-8
Reteach
Transforming Polynomial Functions (continued)
Stretches and compressions are transformations of polynomial functions.
Vertical Stretch or Compression
If f (x) is a polynomial function,
g(x)  af (x) is a vertical stretch or
compression of f (x).
Vertical stretch if a  1
Vertical compression if 0  a  1
Example: f (x)  2x 4  6x 2  4
Vertical compression of f (x)
1
g(x )  f ( x )
2
1
g ( x )  (2 x 4  6 x 2  4)
2
g (x )  2 x 4  6 x 2  4
g (x )  x 4  3 x 2  2
g (x )  x 4  3 x 2 + 2
Horizontal Stretch or Compression
If f (x) is a polynomial function,
1 
g ( x )  f  x  is a horizontal stretch
b 
or compression of f (x).
Horizontal stretch if b  1
Horizontal compression if 0  b  1
Example: f (x)  2x 4  6x 2  4
Horizontal stretch of f(x)
g (x )  2 x 4  6 x 2  4
1 
g(x )  f  x 
2 
4
g(x ) 
2
1 
1 
g(x )  2  x   6  x   4
2 
2 
g(x ) 
1 4 3 2
x  x 4
8
2
1 4 3 2
x  x 4
8
2
Let f (x)  2x 4  6x 2  4. Describe g(x) as a transformation of f (x)
and write the rule for g(x).
3. g(x)  2f (x)
4. g(x)  f (2x)
_________________________________________
________________________________________
_________________________________________
________________________________________
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3-63
Holt McDougal Algebra 2
2. a. g(x) = −2(0.25x)4 + 7(0.25x)2 − 4
4
2
b. g(x) = 4(−2x + 7x − 4)
4. f(x − 2) + 8
5. −f(x + 4) − 3
6. −f(x − 5) − 5
7. f(x + 3) + 4
Problem Solving
1. N(x) = 0.02x3 + 0.4x2 + 0.2x + 235
2. Vertical translation of 200 units up
3.
3. g(x) = −4(x + 2)3 + 5 4. g(x) = 20x3 − 4
5. g(x) = 12(x − 3)3 − 15
4. Because only positive values have
meaning in the context of the problem
6. 6 sin(x) + 1 ≈ 0.3x5 − x3 + 6x + 1
Reteach
5. An additional 200 cars are passing
through the intersection every week.
3
1. Up; g(x) = x + 3
6. Possible answer: R(x) = C(x) − 30;
vertical shift of 30 units down
7. 2C(x) = 0.04x3 + 0.8x2 + 0.4x + 70;
possible answer: a new mall opened at
the intersection.
Reading Strategies
1. a.
3
2. Right; g(x) = (x − 3) + 2
b. (1, −2)
2. a. f(x) = −x3 − 4; f(x) = −x3 + 4
3. Vertical stretch; g(x) = 4x4 − 12x2 + 8
b. g(x) = 6x5 + x3 − 2; g(x) = 6x5 + x3 + 2
4. Horizontal compression; g(x) = 32x4 −
24x2 + 4
c. h(x) = −x2 + 3x − 5; h(x) = x2 + 3x + 5
3-9 CURVE FITTING WITH POLYNOMIAL
Challenge
MODELS
1. f(x) + 11
Practice A
f(x − 2) + 27
−f(x) − 7
2. 0.5f(x)
3. −f(x) − 9
1. First
2. Third
3. Second
4. Second
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A35
Holt McDougal Algebra 2