Chapter 7.6, Problem 5GP

To determine

Find the translation of the given function relates to its parent function.

Answer to Problem 5GP

The graph of $g\left(x\right)=-{\left(\frac{3}{4}\right)}^{x+5}-4$ shows reflection with respect to $x-axis$ , shifted $5-unit$ in the direction of negative $x-axis$ and shifted $4-unit$ in the direction of negative $y-axis$ .

Explanation of Solution

Given information:

It is given in the question that, $g\left(x\right)=-{\left(\frac{3}{4}\right)}^{x+5}-4$

Since $g\left(x\right)=-{\left(\frac{3}{4}\right)}^{x+5}-4$ so let the parent function is $p\left(x\right)={\left(\frac{3}{4}\right)}^x$

By multiplying $"-1"$ , adding $"5"$ in the exponential part and subtracting $"4"$ in non-exponential part of $p\left(x\right)$ then $g\left(x\right)=-{\left(\frac{3}{4}\right)}^{x+5}-4$ 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 $x-axis$ , adding in the exponential part then the graph shifted in the direction of negative $x-axis$ and subtracting in non-exponential part then the graph shifted in the direction of negative $y-axis$ .

So, the graph of $g\left(x\right)=-{\left(\frac{3}{4}\right)}^{x+5}-4$ shows reflection with respect to $x-axis$ , shifted $5-unit$ in the direction of negative $x-axis$ and shifted $4-unit$ in the direction of negative $y-axis$ .