Chapter 4.7, Problem 5E

To determine

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

Answer to Problem 5E

The graph of $g\left(x\right)={\left(x-2\right)}^5-1$ can be found by translating the graph of $f\left(x\right)=x^5$ by $2$ units right horizontally and $1$ unit down vertically.

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^5$ and $g\left(x\right)={\left(x-2\right)}^5-1$

Concept used:

The graph of $g\left(x\right)=f\left(x+h\right)-k$ can be found by translating the graph of $f\left(x\right)$ by $k$ units down vertically and $h$ units left horizontally.

Calculation:

Consider the given functions.

  $f\left(x\right)=x^5$ and $g\left(x\right)={\left(x-2\right)}^5-1$

The function $g\left(x\right)={\left(x-2\right)}^5-1$ is of the form $g\left(x\right)=f\left(x-h\right)+k$ where $h=2,k=-1$ .

So, the graph of $g\left(x\right)={\left(x-2\right)}^5-1$ can be found by translating the graph of $f\left(x\right)=x^5$ by $2$ units right horizontally and $1$ unit down vertically.

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