4.3E: Graphing Polynomial Functions (Exercises)

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Section 4.3 Exercises

In #1-3, determine the end behavior of each polynomial.

1. $f(x)=-3x^4+5x^3-2x+1$

2. $g(x)=2x^7+4x^5-2x^3$

3. $h(x)=-4x^5-7x^8+5x^3-7$

In exercises #4 - 6, match the polynomial function with its graph without using technological assistance.

4. $f(x)=-3x^2-2x$

5. $f(x)=2x^4-50x^2$

6. $f(x)=-2x^3+4x^2-6x$

A. $the graph of a polynomial with one x-intercept that opens down on the right and up on the left.$ B. $The graph of a polynomial with three x-intercepts that opens down on both sides$ C. $The graph of a polynomial with two x-intercepts that opens up on both sides$

D. $the graph of a polynomial with two x-intercepts that opens down on both sides.$ E. $the graph of a polynomial with three x-intercepts that opens down on the right and up on the left.$ F. $the graph of a polynomial with three x-intercepts that opens up on both sides.$

In exercises #7-14, for each polynomial function given: (a) identify each zero and its multiplicity; (b) determine whether the graph crosses the x-axis at each real zero; (c) determine the end behavior; and (d) sketch the graph without using technological assistance.

7. $f(x)=(x+3)^{2} (x-2) = x^3+4x^2-3x-18$

8. $f(x)=-x^3+3x^2 $

9. $f(x)=-x^3+x^2+2x $

10. $f(x)=x^4-4x^2 $

11. $f(x)=-x^4+x^3+6x^2 $

12. $f(x)=x^3-x^2-9x+9 $

13. $f(x)=x^3-x^2-x+1 $

14. $f(x)=x^5-4x^4+4x^3$

In exercises #15-20, write a formula for each polynomial function graphed. Note that the y-intercept that is given on each graph.

15. $The graph of a polynomial with three distinct zeros that opens up on the right and down on the left$ 16. $The graph of a polynomial with three distinct zeros that opens down on the right and up on the left$ 17. $The graph of a polynomial with two distinct zeros that opens down on the right and up on the left$

18. $The graph of a polynomial with two distinct zeros that opens up on the right and down on the left$ 19. $The graph of a polynomial with four distinct zeros that opens down on both sides.$ 20. $The graph of a polynomial with three distinct zeros that opens up on both sides.$

In exercises 21-22, determine whether the statement are true or false. Justify your conclusions.

21. The graph of a polynomial function may not have any x-intercepts.

22. The graph of a polynomial function may not have any y-intercepts.

Answer

1. The graph opens down on the right end (as $x \to \infty$, $y \to -\infty$) and up on the left end (as $x \to - \infty$, $y \to \infty$).

4. Graph D

8. (a) The zeros are $ x=0$ with multiplicity two and $ x= 3$ with multiplicity one. (b) The graph will cross the x-axis at $ x= 3$ but not at $ x= 0$. (c) The graph opens down on the right end (as $x \to \infty$, $y \to -\infty$) and up on the left end (as $x \to - \infty$, $y \to \infty$). (d)

$The graph of a polynomial with two distinct zeros that opens down on the right and up on the left$

15. $ f(x)=\dfrac{1}{2}(x+2)(x-1)(x-3)$

21. True. Since only real zeros of a polynomial are x-intercepts, a polynomial with complex zeros such as $f(x) = x^2+4 $ will have to x-intercepts.

22. False. The y-intercept of a polynomial occurs at $x = 0$, $y=f(0)$. $y=f(0)$ is always defined for polynomials.

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