Chapter 6.4, Problem 45E

To determine

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

Answer to Problem 45E

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

Explanation of Solution

Given information:

Given the function $f\left(x\right)$ and the transformed function $g\left(x\right)$ are given as, $f\left(x\right)=e^x$ and $g\left(x\right)=e^x+4$

Calculation:

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

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

  $f\left(x\right)+4=e^x+4$

  $f\left(x\right)+4=g\left(x\right)$

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

The function $f\left(x\right)=e^x$ has a horizontal asymptote $y=0$ , as the graph of $f\left(x\right)$ is shifted upward by $4$ units, the asymptote of the graph is also shifted upward by 4 units. Then the asymptote to the graph of the function $g\left(x\right)$ is given as $y=4$ .

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

  

Hence, the equation of the asymptote is $y=4$ .