Chapter 4.7, Problem 16E

To determine

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

Answer to Problem 16E

The graph of $g\left(x\right)={\left(2x\right)}^4-1$ can be found by stretching the graph of $f\left(x\right)=x^4$ by a factor of $16$ units vertically followed by a translation of $1$ unit vertically down.

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^4$ and $g\left(x\right)={\left(2x\right)}^4-1$

Concept used:

The graph of $g\left(x\right)=af\left(x\right)-k$ can be found by stretching the graph of $f\left(x\right)$ by a factor of $a$ units vertically followed by a translation of $k$ unit vertically down.

Calculation:

Consider the given functions.

  $f\left(x\right)=x^4$ and $g\left(x\right)={\left(2x\right)}^4-1$

The function $g\left(x\right)={\left(2x\right)}^4-1$ is of the form $g\left(x\right)=af\left(x\right)-k$ where $a=16,k=1$ .

So, the graph of $g\left(x\right)={\left(2x\right)}^4-1$ can be found by stretching the graph of $f\left(x\right)=x^4$ by a factor of $16$ units vertically followed by a translation of $1$ unit vertically down.

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