Chapter 7.6, Problem 13CYU

To determine

Find the translation of the given function.

Answer to Problem 13CYU

The graph of $g\left(x\right)=\frac{2}{3}{\left(5\right)}^{-x}+1$ shows reflection with respect to $y-axis$ , show vertical compress by $factor-\frac{2}{3}$ and graph sifted $1-unit$ in the positive direction of $y-axis$ .

Explanation of Solution

Given information:

It is given in the question that, $g\left(x\right)=\frac{2}{3}{\left(5\right)}^{-x}+1$

Since $g\left(x\right)=\frac{2}{3}{\left(5\right)}^{-x}+1$ so let the parent function be $f\left(x\right)=5^x$

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

So, by the concept of translation of graph when something negative quantity is multiplying in to the exponential part of function then the graph shows reflection with respect to $y-axis$ , when some quantity is multiply to the function which is $">0and<1"$ then the graph show vertical compress and when some quantity is added to the non-exponential part then the graph sifted in the positive direction of $y-axis$ .

So, the graph of $g\left(x\right)=\frac{2}{3}{\left(5\right)}^{-x}+1$ shows reflection with respect to $y-axis$ , show vertical compress by $factor-\frac{2}{3}$ and graph sifted $1-unit$ in the positive direction of $y-axis$ .