Question

$h\left(x\right)=(x+2){(x-4)}^3$

(a) Identify the x-intercepts of the graph of h.

(b) What are the x-intercepts of the graph of $y=h(x-2)$?

Short answer

(a) x-intercepts of$h\left(x\right)$are$-2,4$

(b) x-intercepts of$h(x-2)$are$0,6$

Step-by-step solution

Part (a) Step 1.  Solve for x

Set $y=0$in the given function and solve for $x$

$$\begin{gathered} (x+2){(x-4)}^3=0 \\ x+2=0;x-4=0 \\ x=-2;x=4 \end{gathered}$$

So, the x-intercepts are$-2,4$

Part (b) Step 1. Substitute x with x-2

$$\begin{gathered} h(x-2)=(x-2+2){(x-2-4)}^3 \\ h(x-2)=x{(x-6)}^3 \end{gathered}$$We have,

Part (b) Step 2. Find the x-intercepts of h(x-2)

$$\begin{gathered} x{(x-6)}^3=0 \\ x=0;x-6=0 \\ x=0;x=6 \end{gathered}$$

So, the x-intercepts of$h(x-2)$is$0,6$