Chapter 6.4, Problem 44E

To determine

Write a rule for $g$ that represents the indicated transformation of the graph $f\left(x\right)=8^x$ ; vertical stretch by a factor of $4$ , followed by a translation $1$ units up and 3 units left.

Answer to Problem 44E

The rule for $g$ that represents the indicated transformation of the graph $f$ is $g\left(x\right)=4\cdot 8^{x+3}+1$ .

Explanation of Solution

Given information:

The given function is $f\left(x\right)=8^x$ .

Calculation:

Consider the given function $f\left(x\right)=8^x$ .

First, vertical stretch the function by a factor of $4$ .

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

Therefore, after the vertical stretch, the function becomes $h\left(x\right)=4\cdot 8^x$ .

After that, translate the graph down by $1$ unit and $3$ units left.

For the translating up, add the given unit to the function $f$ and for left translation just add the given unit to the $x$ values of the function.

Therefore, after the translation of the function, it becomes

  $\begin{array}{l}g\left(x\right)=h\left(x+3\right)+1\\ g\left(x\right)=4\cdot 8^{x+3}+1\end{array}$

Therefore, the rule for $g$ that represents the indicated transformation of the graph $f$ is $g(x)=4\cdot 8^{x+3}+1$ .