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 $x→∞,$ $f(x)→∞$

$\star$