3.3e: Exercises - Polynomial End Behaviour

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A: Concepts

Exercise \(\PageIndex{A}\)

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 \(x \rightarrow-\infty, f(x) \rightarrow-\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow-\infty\).

Answers to odd exercises:

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 \(\PageIndex{B}\)

\( \bigstar \) Identify the function as a power function, a polynomial function, or neither. If neither, explain.

5) \(f(x)=(x^2)^3\)

6) \(f(x)=x^5\)

7) \(f(x)=x−x^4\)

8) \(f(x)=\dfrac{x^2}{x^2−1}\)

9) \(f(x)=3^{x+1}\)

10) \(f(x)=2x(x+2)(x−1)^2\)

Answers to odd exercises:: 5. Power function, \(\quad\) 7. Polynomial, \(\quad\) 9. Neither - this is an exponential function because the variable is in the exponent

C: Degree and Leading Coefficient of a Polynomial

Exercise \(\PageIndex{C}\)

\( \bigstar \) (a) Find the degree and leading coefficient for the given polynomial. (b) State the end behaviour.

11) \(7−2x^2\)

12) \(−3x^4\)

13) \(x(4−x^2)(2x+1)\)

14) \(−2x^2− 3x^5+ x−6\)

15) \(x^2(2x−3)^2\)

16) \( -3(2x+5)^2(1-x)^3 \)

Answers to odd exercises:

11. Degree \(2\), Coefficient = \(-2\) \( \swarrow \dots \searrow \)

13. Degree \(4\), Coefficient = \(-2\) \( \swarrow \dots \searrow \)

15. Degree \(4\), Coefficient = \(4\) or \( \nwarrow \dots \nearrow\)

D: Polynomial End Behaviour

Exercise \(\PageIndex{D}\)

\( \bigstar \) Determine the end behavior of the functions.

17) \(f(x)=x^4\)

18) \(f(x)=x^3\)

19) \(f(x)=−x^4\)

20) \(f(x)=−x^9\)

21) \(f(x)=−2x^4− 3x^2+ x−1\)

22) \(f(x)=3x^2+ x−2\)

23) \(f(x)=x^2(2x^3−x+1)\)

24(a) \(f(x)=(2−x)^7\)

24(b) \(f(x)=3x(2x-3)^4\)

25) \(f(x)=x^4−5x^2\)

26) \(f(x)=−x^3\)

27) \(f(x)=(x−1)(x−2)(3−x)\)

28) \(f(x)=x^2(1−x)^2\)

29) \(f(x)=\dfrac{x^5}{10}−x^4\)

Answers to odd exercises:

17. As \(x→\pm ∞\), \(f(x)→∞\), \(\quad\)or \( \nwarrow \dots \nearrow\)

19. As \(x→\pm ∞\), \(f(x)→−∞\), \(\quad\)or \( \swarrow \dots \searrow \)

21. As \(x→\pm∞\), \(f(x)→−∞\), \(\quad\)or \( \swarrow \dots \searrow \)

23. As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞ \)
\(\quad \; \) or \( \swarrow \dots \nearrow \)

25. As \(x→\pm ∞\), \(f(x)→∞\), or \( \nwarrow \dots \nearrow\)

27. as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→−∞\)
\(\quad \; \) or \( \nwarrow \dots \searrow \)

29. As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞ \)
\(\quad \; \) or \( \swarrow \dots \nearrow \)

E: Polynomial Intercepts

Exercise \(\PageIndex{E}\)

\( \bigstar \) 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) \(f(x)=x^4−16\)

34) \(f(x)=x^3+27\)

35) \(f(x)=x(x^2−2x−8)\)

36) \(f(x)=(x+3)(4x^2−1)\)

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) \(C(t)=2t^4−8t^3+6t^2\)

41) \(C(t)=4t^4+12t^3−40t^2\)

42) \(f(x)=x^4−x^2\)

43) \(f(x)=x^3+x^2−20x\)

44) \(f(x)=x^3+6x^2−7x\)

45) \(f(x)=x^3+x^2−4x−4\)

46) \(f(x)=x^3+2x^2−9x−18\)

47) \(f(x)=2x^3−x^2−8x+4\)

48) \(f(x)=x^6−7x^3−8\)

49) \(f(x)=2x^4+6x^2−8\)

50) \(f(x)=x^3−3x^2−x+3\)

51) \(f(x)=x^6−2x^4−3x^2\)

52) \(f(x)=x^6−3x^4−4x^2\)

53) \(f(x)=x^5−5x^3+4x\)

\( \bigstar \) Determine the intercepts and the end behavior.

55) \(f(x)=x^3(x−2)\)

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) \(f(x)=x^3−16x\)

60) \(f(x)=x^3−27\)

61) \(f(x)=x^4−81\)

62) \(f(x)=−x^3+x^2+2x\)

63) \(f(x)=x^3−2x^2−15x\)

64) \(f(x)=x^3−0.01x\)

Answers to odd exercises:

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. \((0, 4)\); \((−2,0),\;(2,0),\left(\tfrac{1}{2},0\right)\)

49. \((0, -8)\); \((1,0),\;(−1,0)\)

51. \((0,0),\;(\sqrt{3},0)),\;(−\sqrt{3},0)\)

53. \((0,0),\,(1,0),(−1,0),(2,0),(−2,0)\)

55. \((0, 0)\); \((0, 0), (2, 0)\). As \(x→\pm ∞,\) \(f(x)→∞\)

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 \(x→\pm ∞,\) \(f(x)→∞\)

63.\((0, 0)\); \((−3, 0), (0, 0), (5, 0).\) As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

\( \star \)