Chapter 7.6, Problem 31PPS

To determine

Find the translation of the function.

Answer to Problem 31PPS

The graph of $g\left(x\right)=-\frac{1}{3}{\left(2\right)}^{x+3}$ shows reflection with respect to $x-axis$ , shifted $3-unit$ with respect to negative $x-axis$ and the graph shows vertical compress by $factor-\frac{1}{3}.$

Explanation of Solution

Given information:

It is given in the question that, $g\left(x\right)=-\frac{1}{3}{\left(2\right)}^{x+3}$

Since, $g\left(x\right)=-\frac{1}{3}{\left(2\right)}^{x+3}$ so let the parent function be $f\left(x\right)={\left(2\right)}^x$

By multiplying $f\left(x\right)$ to $"-\frac{1}{3}"$ and adding $"3"$ in the exponential part then $g\left(x\right)=-\frac{1}{3}{\left(2\right)}^{x+3}$ is arrived

So, by the concept of translation of graph when something negative quantity is multiplying in the function then the graph shows reflection with respect to $x-axis$ , when some quantity is adding in the exponential part of function then the graph shifted with respect to negative $x-axis$ and when some quantity which is $>0and<1$ is multiply to the function then the graph shows vertical compress.

So, the graph of $g\left(x\right)=-\frac{1}{3}{\left(2\right)}^{x+3}$ shows reflection with respect to $x-axis$ , shifted $3-unit$ with respect to negative $x-axis$ and the graph shows vertical compress by $factor-\frac{1}{3}.$