Chapter 6.4, Problem 35E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)=5^x$ ; translation $2$ units down, followed by reflection in the $x$ -axis.

Answer to Problem 35E

The rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=5^{-x}-2$ .

Explanation of Solution

Given information:

The given function $f\left(x\right)=5^x$.

Calculation:

  $f\left(x\right)=5^x$

First, translate the function2 units down.

For translation of any function down, subtract $2$ units from the $y-$ coordinates in the given function.

Therefore, after the translation the function becomes $f\left(x\right)=5^x-2$.

After that,reflect it in the y-axis.

For the reflection of any function $f\left(x\right)$ , just change the sign of the value of $x$ that is $f\left(-x\right)$ .

  $f\left(x\right)=f\left(-x\right)$

  $f\left(x\right)=5^{-x}-2$

Therefore, the rule of the transformed function $g\left(x\right)$ of the given function $f$ is $g\left(x\right)=5^{-x}-2$ .

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=5^{-x}-2$ .