Chapter 4.7, Problem 15E

To determine

To describe: the transformation of $f$ represented by $g$ . Graph the functions $f$ and $g$ .

Answer to Problem 15E

The graph of $g\left(x\right)=\frac{3}{4}{\left(x+4\right)}^5$ can be found by shrunken the graph of $f\left(x\right)=x^5$ by a factor of $\frac{3}{4}$ units vertically followed by a translation of $4$ unit horizontally left.

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^5$ and $g\left(x\right)=\frac{3}{4}{\left(x+4\right)}^5$

Concept used:

The graph of $g\left(x\right)=\frac{1}{a}f\left(x+h\right)$ can be found by shrunken the graph of $f\left(x\right)$ by a factor of $a$ units vertically followed by a translation of $h$ unit horizontally left.

Calculation:

Consider the given functions.

  $f\left(x\right)=x^5$ and $g\left(x\right)=\frac{3}{4}{\left(x+4\right)}^5$

The function $g\left(x\right)=\frac{3}{4}{\left(x+4\right)}^5$ is of the form $g\left(x\right)=\frac{1}{a}f\left(x+h\right)$ where $a=\frac{4}{3},h=4$ .

So, the graph of $g\left(x\right)=\frac{3}{4}{\left(x+4\right)}^5$ can be found by shrunken the graph of $f\left(x\right)=x^5$ by a factor of $\frac{3}{4}$ units vertically followed by a translation of $4$ unit horizontally left.

Now, sketch both the graphs in same set of axes as shown: