Name ________________________________________ Date __________________ Class__________________
LESSON
3-7
Reteach
Investigating Graphs of Polynomial Functions
Examine the sign and the exponent of the leading term (term of greatest degree) of a
polynomial P(x) to determine the end behavior of the function.
Even degree functions: Exponent of leading term is even.
Read:
Positive leading coefficient
Negative leading coefficient
x
approaches
As
As x  , P(x)  .
As x  , P(x)  .
positive infinity, P(x)
As x  , P(x)  .
As x  , P(x)  .
approaches negative
infinity.
Example: P(x)  3x 4  2x 3  5
End behavior:
Leading term: 3x 4
As x  , P(x)  .
As x  , P(x)  .
Odd degree functions: Exponent of leading term is odd.
Positive leading coefficient
As x  , P(x)  .
As x  , P(x)  .
Sign: positive
Degree: 4, even
Negative leading coefficient
As x  , P(x)  .
As x  , P(x)  .
Example: P(x)  2x 5  6x 2  x Leading term: 2x 5
End behavior: As x  , P(x)  .
As x  , P(x)  .
Sign: negative
Degree: 5, odd
Identify the end behavior of each function.
1. P(x)  4x 3  8x 2  5
Leading term: 4x 3
Sign and degree: ________________
End behavior: ___________________
2. P(x)  9x 6  2x 3  x  7
Leading term: _______________
Sign and degree: ______________
End behavior:___________________
_________________________________________
________________________________________
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3-54
Holt McDougal Algebra 2
Name ________________________________________ Date __________________ Class__________________
LESSON
3-7
Reteach
Investigating Graphs of Polynomial Functions (continued)
You can use the graph of a polynomial function to analyze the function.
P(x): odd degree and
positive leading coefficient if:
Notice the graph
increases, then
decreases, and then
increases again.
As x  , P(x)   and
as x  , P(x) x  .
P(x): odd degree and
negative leading coefficient if:
Notice this graph is the
reverse. It decreases,
then increases, and
then decreases again.
As x  , P(x) x   and
as x  , P(x) x  .
P(x): even degree and
positive leading coefficient if:
Look at the end
behavior. The graph
increases at both ends.
As x  , P(x) x   and
as x  , P(x) x  .
P(x): even degree and
negative leading coefficient if:
This is the reverse. The
graph decreases at both
ends.
As x  , P(x)   and
as x  , P(x)  .
Identify whether each function has an odd or even degree and a
positive or negative leading coefficient.
3.
4.
5.
________________________
_________________________
________________________
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3-55
Holt McDougal Algebra 2
9. y = x4; 1; looks like a parabola
7. Minima: 4.5; maxima: 5.1 and 13.5
8. Minima: −8.68; maxima: 0
Problem Solving
9. a. 3.03 m3
1. V(x) = x(11 − 2x)(17 − 2x)
b. 1.9 m by 2.9 m by 0.55 m
2. a. 4x3 − 56x2 + 187x = 0
b. Positive
Reteach
1. Positive, 3, odd
c. Odd
as x → −∞, P(x) → −∞
d. As x → +∞, V → +∞, and as x → −∞,
V → −∞.
as x → +∞, P(x) → +∞
3. a. 2
2. −9x6
b. About 183 and −64
negative, 6, even
4. Values of x greater than 5.5 or less than
0
as x → −∞, P(x) → −∞
as x → +∞, P(x) → −∞
3. Even; positive
5. About 2.3 and 183
inches
4. Odd; negative
5. Odd positive
6. 183 cubic
7. 2.3 in. by 6.4 in. by 12.4 in.
8. A
Challenge
1. a. 3
9. D
Reading Strategies
b. y = (x + 3)(x + 1)(x − 1)(x − 5)
1. a. Its degree is even and the leading
coefficient is less than zero.
c.
b. The function approaches −∞ as x →
+∞.
c. The function approaches −∞ as x →
−∞.
2. As x → +∞, P(x) → +∞; As x → −∞, P(x)
→ −∞
3. Odd
4. It is positive.
3-8 TRANSFORMING POLYNOMIAL
FUNCTIONS
2. y = (x + 3)(x + 1) (x − 1)(x − 5); 3;
4 distinct x- intercepts
Practice A
2
3. y = (x + 3) (x − 1) (x − 5); 3; tangent to
the x-axis at x = −3
1. Translated 5 units up
4. y = (x + 3)3(x − 5); 1; curve “bends” at
x = −3
3. Translated 1 unit right
2. Translated 10 units left
4. Translated 6 units down
5. y = (x + 3)4; 1; tangent to the x-axis at
x = −3
5. g(x) = −x3 − 2x − 3
6. y = x (x + 3) (x − 1)(x − 5); 3; 4 distinct
x- intercepts
6. g(x) = −x3 + x + 1
7. y = x2(x + 3) (x − 3); 3; tangent to the
x-axis at x = 0
8. g(x) =
7. g(x) =
1 4
x +1
2
1 3
1
x + x2 − 6
27
9
8. y = x3(x − 3); 1; curve “bends” at x = 0
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Holt McDougal Algebra 2