ID: 1
Algebra 2
Name___________________________________
Polynomials - End Behavior
Date________________ Period____
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Describe the end behavior of each function.
1) f ( x) = x 3 + 10 x 2 + 32 x + 34
2) f ( x) = − x 2 − 8 x − 15
3) f ( x) = − x 4 + x 2 + 2
4) f ( x) = x 4 − 4 x 2 − x + 3
5) f ( x) = − x 3 + 2 x 2 + 2
6) f ( x) = x 4 − x 2 − 2
7) f ( x) = x 3 − 3 x 2 + 1
8) f ( x) = x 5 − 4 x 3 + x + 1
9) f ( x) = − x 5 + 4 x 3 − 5 x − 4
10) f ( x) = − x 3 + 3 x 2 − 4
11) f ( x) = x 4 − 3 x 2 − 3 x + 4
12) f ( x) = − x 5 + 4 x 3 − 2 x − 2
13) f ( x) = x 4 − 4 x 2 − x + 5
Sketch the general shape of each function.
14) f ( x) = x 3 − x 2 + 4
15) f ( x) = x 5 − 3 x 3 + 2 x + 4
16) f ( x) = − x 3 + x 2 − 1
17) f ( x) = − x 4 + 3 x 2 − 2 x − 4
18) f ( x) = 2 x 2 − 3
State the maximum number of turns the graph of each function could make. Then sketch the graph.
19) f ( x) = − x 3 + 3 x 2 − 2
20) f ( x) = x 3 − 3 x 2 + 4
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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-1-
4
6
8 x
Worksheet by Kuta Software LLC
21) f ( x) = x 3 − 10 x 2 + 33 x − 38
22) f ( x) = − x 3 + x 2 + 1
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
23) f ( x) = − x 3 − 7 x 2 − 15 x − 8
−6
−4
8
6
6
4
4
2
2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
25) f ( x) = x 4 − 2 x 2 + x − 2
−6
−4
6
8 x
6
8 x
y
8
8
6
6
4
4
2
2
−2
4
26) f ( x) = − x 3 + 2 x 2 + 1
y
−8
8 x
y
8
−2
6
24) f ( x) = − x 5 + 2 x 3 − x − 4
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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-2-
4
Worksheet by Kuta Software LLC
Sketch the graph of each function. Approximate the relative minima and relative maxima to the nearest
tenth.
27) f ( x) = − x 3 − 6 x 2 − 9 x − 4
28) f ( x) = − x 3 + 4 x 2 − 7
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
29) f ( x) = x 3 − 4 x 2 + 3
−6
−4
8 x
4
6
8 x
y
8
8
6
6
4
4
2
2
−2
6
30) f ( x) = x 3 − 4 x 2 + 2
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
31) f ( x) = x 3 − 4 x 2 + 7
y
8
6
4
2
−8
−6
−4
−2
2
4
6
8 x
−2
−4
−6
−8
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-3-
Worksheet by Kuta Software LLC
Sketch the graph of each function. Approximate each real zero to the nearest tenth. Approximate the
relative minima and relative maxima to the nearest tenth.
32) f ( x) = x 4 − 3 x 2 + 3 x − 1
33) f ( x) = − x 4 + 4 x 3 − 4 x 2 − 1
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
34) f ( x) = − x 5 + 4 x 3 − x − 1
−6
−4
8 x
y
8
8
6
6
4
4
2
2
−2
6
35) f ( x) = x 5 − 4 x 3 + 5 x + 4
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
36) f ( x) = x 5 − 4 x 3 + x
y
8
6
4
2
−8
−6
−4
−2
2
4
6
8 x
−2
−4
−6
−8
Name each polynomial by degree and number of terms.
37) x 8 + 4 + 6 x 3
38) −4 + 5m
39) 6n 4 + 10 − 9n 2 − 9n 3
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-4-
Worksheet by Kuta Software LLC
ID: 1
Algebra 2
Name___________________________________
Polynomials - End Behavior
Date________________ Period____
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Describe the end behavior of each function.
1) f ( x) = x 3 + 10 x 2 + 32 x + 34
f ( x) → −∞ as x → −∞2) f ( x) = − x 2 − 8 x − 15 f ( x) → −∞ as x → −∞
f ( x) → +∞ as x → +∞
f ( x) → −∞ as x → +∞
4
2
4
2
3) f ( x) = − x + x + 2 f ( x) → −∞ as x → −∞
4) f ( x) = x − 4 x − x + 3 f ( x) → +∞ as x → −∞
f ( x) → −∞ as x → +∞
f ( x) → +∞ as x → +∞
3
2
4
2
5) f ( x) = − x + 2 x + 2 f ( x) → +∞ as x → −∞
6) f ( x) = x − x − 2 f ( x) → +∞ as x → −∞
f ( x) → −∞ as x → +∞
f ( x) → +∞ as x → +∞
3
2
5
3
(
)
(
)
(
)
7) f x = x − 3 x + 1 f x → −∞ as x → −∞
8) f x = x − 4 x + x + 1 f ( x) → −∞ as x → −∞
f ( x) → +∞ as x → +∞
f ( x) → +∞ as x → +∞
5
3
3
2
9) f ( x) = − x + 4 x − 5 x − 4 f ( x) → +∞ as x → −∞ 10) f ( x) = − x + 3 x − 4 f ( x) → +∞ as x → −∞
f ( x) → −∞ as x → +∞
f ( x) → −∞ as x → +∞
4
2
5
3
11) f ( x) = x − 3 x − 3 x + 4 f ( x) → +∞ as x → −∞ 12) f ( x) = − x + 4 x − 2 x − 2 f ( x) → +∞ as x → −∞
f ( x) → +∞ as x → +∞
f ( x) → −∞ as x → +∞
4
2
(
)
(
)
13) f x = x − 4 x − x + 5 f x → +∞ as x → −∞
f ( x) → +∞ as x → +∞
Sketch the general shape of each function.
14) f ( x) = x 3 − x 2 + 4
15) f ( x) = x 5 − 3 x 3 + 2 x + 4
y
16) f ( x) = − x 3 + x 2 − 1
17) f ( x) = − x 4 + 3 x 2 − 2 x − 4
y
18) f ( x) = 2 x 2 − 3
y
y
x
y
x
State the maximum number of turns the graph of each function could make. Then sketch the graph.
x
19) f ( x) = − x + 3 x − 2
3
3
y
−6
−4
2
x
y
Max # Turns: 2
8
−8
6
4
4
2
2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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Max # Turns: 2
8
6
−2
x
20) f ( x) = x − 3 x + 4
2
-1-
4
6
8 x
Worksheet by Kuta Software LLC
21) f ( x) = x 3 − 10 x 2 + 33 x − 38
22) f ( x) = − x 3 + x 2 + 1
y
−8
−6
−4
6
4
4
2
2
−2
2
4
6
8 x
−8
−4
−6
−6
−8
−8
4
4
2
2
4
6
8 x
−8
−6
−4
−2
−2
−4
−4
−6
−6
−8
−8
8 x
y
6
4
4
2
2
6
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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Max # Turns: 2
8
6
4
4
26) f ( x) = − x 3 + 2 x 2 + 1
Max # Turns: 3
2
Max # Turns: 4
2
−2
−2
8 x
8
6
2
6
y
Max # Turns: 2
6
−2
4
24) f ( x) = − x 5 + 2 x 3 − x − 4
y
−4
2
−4
8
−6
−2
−4
25) f ( x) = x 4 − 2 x 2 + x − 2
−8
−4
−2
y
−6
−6
−2
8
Max # Turns: 2
8
6
23) f ( x) = − x 3 − 7 x 2 − 15 x − 8
−8
y
Max # Turns: 2
8
-2-
4
6
8 x
Worksheet by Kuta Software LLC
Sketch the graph of each function. Approximate the relative minima and relative maxima to the nearest
tenth.
27) f ( x) = − x 3 − 6 x 2 − 9 x − 4
28) f ( x) = − x 3 + 4 x 2 − 7
y
−8
−6
−4
6
4
4
2
2
−2
2
4
6
8 x
−8
−4
−2
2
−2
−4
−4
−6
−6
−8
−8
8 x
4
4
2
2
4
6
8 x
Minima: (2.7, −7.5)
Maxima: (0, 2)
8
6
2
6
y
Minima: (2.7, −6.5)
Maxima: (0, 3)
6
−2
4
30) f ( x) = x 3 − 4 x 2 + 2
y
−4
−6
−2
8
−6
Minima: (0, −7)
Maxima: (2.7, 2.5)
8
6
29) f ( x) = x 3 − 4 x 2 + 3
−8
y
Minima: (−3, −4)
Maxima: (−1, 0)
8
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
31) f ( x) = x 3 − 4 x 2 + 7
y
Minima: (2.7, −2.5)
Maxima: (0, 7)
8
6
4
2
−8
−6
−4
−2
2
4
6
8 x
−2
−4
−6
−8
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-3-
Worksheet by Kuta Software LLC
Sketch the graph of each function. Approximate each real zero to the nearest tenth. Approximate the
relative minima and relative maxima to the nearest tenth.
32) f ( x) = x 4 − 3 x 2 + 3 x − 1
33) f ( x) = − x 4 + 4 x 3 − 4 x 2 − 1
y
6
−8
−6
−4
y
Real Zeros: −2.1, 1
Minima: (−1.4, −7.2)
Maxima: None
8
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
−2
−4
−4
−6
−6
−8
−8
y
4
8 x
Real Zeros: −1.8
Minima: (−0.7, 1.7)
(1.4, 5.4)
Maxima: (−1.4, 2.6)
(0.7, 6.3)
8
6
4
2
−4
6
y
Real Zeros: −2, 0.8, 1.9
Minima: (−1.5, −5.4)
(0.3, −1.2)
Maxima: (−0.3, −0.8)
(1.5, 3.4)
6
−6
4
35) f ( x) = x 5 − 4 x 3 + 5 x + 4
8
−8
2
−2
34) f ( x) = − x 5 + 4 x 3 − x − 1
Real Zeros: None
Minima: (1, −2)
Maxima: (0, −1)
(2, −1)
8
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
36) f ( x) = x 5 − 4 x 3 + x
y
Real Zeros: −1.9, −0.5, 0.5, 0, 1.9
Minima: (−0.3, −0.2)
(1.5, −4.4)
Maxima: (−1.5, 4.4)
(0.3, 0.2)
8
6
4
2
−8
−6
−4
−2
2
4
6
8 x
−2
−4
−6
−8
Name each polynomial by degree and number of terms.
37) x 8 + 4 + 6 x 3
eighth degree trinomial
39) 6n 4 + 10 − 9n 2 − 9n 3
quartic polynomial with four terms
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38) −4 + 5m
linear binomial
-4-
Worksheet by Kuta Software LLC