Practice 1 Solutions

1. Explain when a polynomial function is also a power function.
   

A polynomial function is also a power function when the equation contains a monomial term raised to a whole number exponent.

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

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

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

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.

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

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

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

This is not a polynomial function because it is not smooth (there are sharp peaks).

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

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

10. $g\left(x\right)=7x^2–2x+5$

$a=7, n=2$

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

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

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

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

12. $y=\frac{4}{3}{\left(x–8\right)}^3$

$a=\frac{4}{3}, n=3$

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

13. $h\left(x\right)=–2x{\left(x+6\right)}^7{\left(x–2\right)}^2$

$$\begin{gathered} n=1+7+2 \\ a=–2, n=10 \end{gathered}$$

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

14. Draw a Venn diagram to sort the equations into polynomial function, power function, both, or neither.
    $r\left(x\right)=x^{–\frac{1}{5}} y=2^x+1 f\left(x\right)=–{\left(5x–1\right)}^3{\left(x+2\right)}^2 m\left(x\right)=\frac{5}{8}x–6 A=\mathrm{\pi }{r}^2$

r(x): Power, monomial with a rational exponent (negative fraction)

y: Neither, the exponent is a variable

f(x): Polynomial function, more than one term raised to a whole number exponent

m(x): Polynomial function, more than one term raised to a whole number exponent

A: Polynomial and power function, monomial with a whole number exponent