3.3e: Exercises - Polynomial End Behaviour

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 \(x→∞,\) \(f(x)→∞\)

\( \star \)