Chapter 7.6, Problem 15CYU

To determine

Find the translation of the function.

Answer to Problem 15CYU

The graph of $g\left(x\right)={\left(\frac{5}{4}\right)}^{4x}-2$ bent towards the $y-axis$ and shifted $2-unit$ in the negative direction of $y-axis$ .

Explanation of Solution

Given information:

It is given in the question that, $g\left(x\right)={\left(\frac{5}{4}\right)}^{4x}-2$

Since $g\left(x\right)={\left(\frac{5}{4}\right)}^{4x}-2$ so let the parent function be $f\left(x\right)={\left(\frac{5}{4}\right)}^x$

By multiplying $"4"$ to the exponential part of $f\left(x\right)$ and subtracting $2-unit$ in to the non-exponential part of function then $g\left(x\right)={\left(\frac{5}{4}\right)}^{4x}-2$ is arrived

So, by the concept of translation of graph when something positive quantity is multiplying in to the exponential part of function then the graph bent towards the $y-axis$ .and when some quantity is subtracted in to the non-exponential part of function then the graph shifted in the negative direction of $y-axis$

So, the graph of $g\left(x\right)={\left(\frac{5}{4}\right)}^{4x}-2$ bent towards the $y-axis$ and graph shifted $2-unit$ in the negative direction of $y-axis$