Chapter 7.6, Problem 34PPS

To determine

Find the translation of the given function.

Answer to Problem 34PPS

The graph of $g\left(x\right)=-8{\left(\frac{2}{3}\right)}^{x+3}-4$ shows reflection with respect to $x-axis$ , the graph show vertical stretch by $factor-8$ , the graph shifted $3-unit$ in the negative $x-axis$ and graph shifted $4-unit$ in the negative $y-axis$ .

Explanation of Solution

Given information:

It is given in the question that, $g\left(x\right)=-8{\left(\frac{2}{3}\right)}^{x+3}-4$

By multiplying $f\left(x\right)$ to $"-8"$ and adding $"3"$ in the exponential part and subtracting $"4"$ in the non-exponential part then $g\left(x\right)=-8{\left(\frac{2}{3}\right)}^{x+3}-4$ is arrived

So, by the concept of translation of graph when $"-1"$ is multiplying in the function then the graph shows reflection with respect to $x-axis$ ,when $"8"$ multiply to the function then the graph shows vertical stretch by $factor-8$ ,by adding $"3"$ in the exponential part of the function then the graph shifted in the negative $x-axis$ and by subtracting $"4"$ in the non-exponential part of the function then the graph shifted in the negative $y-axis$

So, the graph of $g\left(x\right)=-8{\left(\frac{2}{3}\right)}^{x+3}-4$ shows reflection with respect to $x-axis$ , the graph show vertical stretch by $factor-8$ , the graph shifted $3-unit$ in the negative $x-axis$ and graph shifted $4-unit$ in the negative $y-axis$ .