Practice 2 Solutions

This is a polynomial function because it is a smooth, continuous graph.

$$\begin{gathered} a=\mathrm{negative}, n=\mathrm{odd} \\ x\to –\infty , f\left(x\right)\to +\infty , \mathrm{and} x\to +\infty , f\left(x\right)\to –\infty \end{gathered}$$

This is a polynomial function because it is a smooth, continuous graph.

$$\begin{gathered} a=\mathrm{postive}, n=\mathrm{odd} \\ x\to –\infty , f\left(x\right)\to –\infty , \mathrm{and} x\to +\infty , f\left(x\right)\to +\infty \end{gathered}$$

This is a polynomial function because it is a smooth, continuous graph.

$$\begin{gathered} a=\mathrm{negative}, n=\mathrm{even} \\ x\to \pm \infty , f\left(x\right)\to –\infty \end{gathered}$$

This is not a polynomial function because it is not continuous (there is a break in the graph).

5. $f\left(x\right)={\left(3x–1\right)}^4$

Polynomial

The coefficients are real numbers and the exponents are whole numbers.

6. $q\left(n\right)=5{\left(n–8\right)}^{–3}+n$

Neither

There is more than one term (not a power function), and the exponent is a negative number.

7. $j\left(x\right)=–8x^{\frac{1}{3}}$

Power

There is one term raised to a fixed fractional power.

8. $y=3x^5–16x^3+x^2+x–14$

Polynomial

The coefficients are real numbers, and the exponents are whole numbers.

9. $q\left(x\right)=5{\left(x–8\right)}^7+x$

$a=5, n=7$

$x\to –\infty , f\left(x\right)\to –\infty , \mathrm{and} x\to +\infty , f\left(x\right)\to +\infty$

10. $f\left(x\right)=–x^4–3x^3+2x^2+8x–1$

$a=–1, n=4$

$x\to \pm \infty , f\left(x\right)\to –\infty$

11. $h\left(x\right)=–18x^3\left(x–1\right)\left(x+7\right)$

$$\begin{gathered} n=3+1+1 \\ a=–18, n=5 \end{gathered}$$

$x\to –\infty , f\left(x\right)\to +\infty , \mathrm{and} x\to +\infty , f\left(x\right)\to –\infty$

12. $b\left(x\right)=x^2{\left(x+7\right)}^3\left(x–6\right)$

$$\begin{gathered} n=2+3+1 \\ a=1, n=6 \end{gathered}$$

$x\to \pm \infty , f\left(x\right)\to +\infty$

14. Explain what is needed to determine the end behavior of a polynomial function from an equation.

The leading coefficient and the degree of the polynomial (the leading coefficient test) determine the end behavior of a polynomial function.