Chapter 6.4, Problem 46E

To determine

Describethe transformation of the graph $f$ and then give the equation of the asymptote.

Answer to Problem 46E

The graph of the function $f\left(x\right)$ is shifted right by 4 units. The required equation of the asymptote is $y=0$ .

Explanation of Solution

Given information:

The given the function $f\left(x\right)$ and the transformed function $g\left(x\right)$ are given as $f\left(x\right)=3^x$ and $g\left(x\right)=3^{x-4}$.

Calculation:

Consider the given function $f\left(x\right)$ as $f\left(x\right)=3^x$.

It can be seen that the transformed graph $g\left(x\right)$ can be obtained by subtracting $4$ in the $x-$ coordinate of the function $f\left(x\right)$ .

  $f\left(x-3\right)=3^{x-4}$

  $f\left(x-3\right)=g\left(x\right)$

On subtract 4 in the $x-$ coordinate of the function of $f\left(x\right)$ , the graph of $f\left(x\right)$ is shifted right by $4$ units.

The function $f\left(x\right)=3^x$ has a horizontal asymptote $y=0$ , as the graph of $f\left(x\right)$ is shifted right side by $4$ units, the asymptote of the graph has no change due to no change in the $y-$ coordinate. Then the asymptote to the graph of the function $g\left(x\right)$ is given as $y=0$ .

The graph of the function $g\left(x\right)$ and its asymptote is given below.

  

Hence, the equation of the asymptote is $y=0$ .