3.3e: Exercises - Polynomial End Behaviour
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A: Concepts
Exercise 3.3e.A
1) Explain the difference between the coefficient of a power function and its degree.
2) If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?
3) In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.
4) What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x→−∞,f(x)→−∞ and as x→∞,f(x)→−∞.
- Answers to odd exercises:
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1. The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
3. As x decreases without bound, so does f(x). As x increases without bound, so does f(x).
B: Identify Power Functions and Polynomials
Exercise 3.3e.B
★ Identify the function as a power function, a polynomial function, or neither. If neither, explain.
5) f(x)=(x2)3 6) f(x)=x5 |
7) f(x)=x−x4 8) f(x)=x2x2−1 |
9) f(x)=3x+1 10) f(x)=2x(x+2)(x−1)2 |
- Answers to odd exercises:
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5. Power function, 7. Polynomial, 9. Neither - this is an exponential function because the variable is in the exponent
C: Degree and Leading Coefficient of a Polynomial
Exercise 3.3e.C
★ (a) Find the degree and leading coefficient for the given polynomial. (b) State the end behaviour.
11) 7−2x2 12) −3x4 |
13) x(4−x2)(2x+1) 14) −2x2−3x5+x−6 |
15) x2(2x−3)2 16) −3(2x+5)2(1−x)3 |
- Answers to odd exercises:
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11. Degree 2, Coefficient = −2 ↙⋯↘ 13. Degree 4, Coefficient = −2 ↙⋯↘ 15. Degree 4, Coefficient = 4 or ↖⋯↗
D: Polynomial End Behaviour
Exercise 3.3e.D
★ Determine the end behavior of the functions.
17) f(x)=x4 18) f(x)=x3 19) f(x)=−x4 20) f(x)=−x9
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21) f(x)=−2x4−3x2+x−1 22) f(x)=3x2+x−2 23) f(x)=x2(2x3−x+1) 24(a) f(x)=(2−x)7 24(b) f(x)=3x(2x−3)4 |
25) f(x)=x4−5x2 26) f(x)=−x3 27) f(x)=(x−1)(x−2)(3−x) 28) f(x)=x2(1−x)2 29) f(x)=x510−x4 |
- Answers to odd exercises:
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17. As x→±∞, f(x)→∞, or ↖⋯↗
19. As x→±∞, f(x)→−∞, or ↙⋯↘
21. As x→±∞, f(x)→−∞, or ↙⋯↘
23. As x→∞, f(x)→∞, as x→−∞, f(x)→−∞
or ↙⋯↗25. As x→±∞, f(x)→∞, or ↖⋯↗
27. as x→−∞, f(x)→∞, as x→∞, f(x)→−∞
or ↖⋯↘29. As x→∞, f(x)→∞, as x→−∞, f(x)→−∞
or ↙⋯↗
E: Polynomial Intercepts
Exercise 3.3e.E
★ Find the intercepts of the functions.
30) C(t)=2(t−4)(t+1)(t−6) 31) f(t)=2(t−1)(t+2)(t−3) 32) g(n)=−2(3n−1)(2n+1) 33) f(x)=x4−16 34) f(x)=x3+27 35) f(x)=x(x2−2x−8) 36) f(x)=(x+3)(4x2−1) 37) C(t)=3(t+2)(t−3)(t+5) |
38) C(t)=4t(t−2)2(t+1) 39) C(t)=2t(t−3)(t+1)2 40) C(t)=2t4−8t3+6t2 41) C(t)=4t4+12t3−40t2 42) f(x)=x4−x2 43) f(x)=x3+x2−20x 44) f(x)=x3+6x2−7x 45) f(x)=x3+x2−4x−4 |
46) f(x)=x3+2x2−9x−18 47) f(x)=2x3−x2−8x+4 48) f(x)=x6−7x3−8 49) f(x)=2x4+6x2−8 50) f(x)=x3−3x2−x+3 51) f(x)=x6−2x4−3x2 52) f(x)=x6−3x4−4x2 53) f(x)=x5−5x3+4x |
★ Determine the intercepts and the end behavior.
55) f(x)=x3(x−2) 56) f(x)=x(x−3)(x+3) 57) f(x)=x(14−2x)(10−2x) 58) f(x)=x(14−2x)(10−2x)2 |
59) f(x)=x3−16x 60) f(x)=x3−27 61) f(x)=x4−81 62) f(x)=−x3+x2+2x |
63) f(x)=x3−2x2−15x 64) f(x)=x3−0.01x |
- Answers to odd exercises:
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31.(0,12); (1,0),(–2,0),(3,0)
33. (0,−16); (2,0),(−2,0).
35. (0,0),(4,0),(−2,0).
37. (0,90); (−2,0),(3,0),(−5,0)
39. (0,0), (3,0),(−1,0)
41. (0,0),(−5,0),(2,0)
43. (0,0),(−5,0),(4,0)
45. (0,−4); (2,0),(−2,0),(−1,0)
47. (0,4); (−2,0),(2,0),(12,0)
49. (0,−8); (1,0),(−1,0)
51. (0,0),(√3,0)),(−√3,0)
53. (0,0),(1,0),(−1,0),(2,0),(−2,0)
55. (0,0); (0,0),(2,0). As x→±∞, f(x)→∞
57. (0,0); (0,0),(5,0),(7,0). As x→−∞, f(x)→−∞, as x→∞, f(x)→∞
59. (0,0); (−4,0),(0,0),(4,0). As x→−∞, f(x)→−∞, as x→∞, f(x)→∞
61. (0,−81); (3,0),(−3,0). As x→±∞, f(x)→∞
63.(0,0); (−3,0),(0,0),(5,0). As x→−∞, f(x)→−∞, as x→∞, f(x)→∞
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