Chapter 6.4, Problem 38E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)=e^{-x}$ ; translate $4$ units right and $1$ unit down, followed by a vertical shrink by a factor of $\frac{1}{3}$ .

Answer to Problem 38E

The rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=\frac{1}{3}\left(e^{-x-4}-1\right)$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=e^{-x}$ .

Calculation:

Consider the given function

  $f\left(x\right)=e^{-x}$ .

First, translate the function $f$ by $4$ units right.

For translate the function right by some units, subtract $x$ value of the function by the given unit.

Therefore, after the translation, the function becomes $f\left(x\right)=e^{-x-4}$ .

After that, translate the graph down by $1$ unit.

For the translation by shiftingdown, subtractgiven unit to the function $f$ .

Therefore after the translation, the function becomes $f\left(x\right)=e^{-x-4}-1$ .

Finally, vertically shrink the function by a factor of $\frac{1}{3}$ .

For the shrinking of function, multiply the function with $\frac{1}{givenunit}$ .

Therefore after the shrinking of function, the obtained function is $f\left(x\right)=\frac{1}{3}\left(e^{-x-4}-1\right)$ .

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=\frac{1}{3}\left(e^{-x-4}-1\right)$ .