Chapter 6.4, Problem 39E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)={\log }_{6}x$ ; vertical stretch by a factor of $6$ , followed by a translation $5$ units down.

Answer to Problem 39E

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

Explanation of Solution

Given information:

The given function is $f\left(x\right)={\log }_{6}x$ .

Calculation:

Consider the given function $f\left(x\right)={\log }_{6}x$ .

First, vertical stretch the function by a factor of $6$ .

For translate the vertical stretch, multiply the function $f$ with given unit.

Therefore, after the vertical stretch, the function becomes

  $f\left(x\right)=6{\log }_{6}x$ .

Substitute $6{\log }_6=1$ into the above equation.

  $f\left(x\right)=x$

After that, translate the graph downward by $5$ unit.

For the translating down, subtract the given unit to the function $f$ .

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

Hence, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g(x)=x-5$ .