Chapter 4.7, Problem 22E

To determine

To find: the correction of error in describing the transformation of the graph of function represented by another function.

Answer to Problem 22E

The graph of the function $g\left(x\right)$ is vertically stretched by the factor of $243$ and vertical shifted of $4$ unit down.

Explanation of Solution

Given:

  $f\left(x\right)=x^5,g\left(x\right)={\left(3w\right)}^5-4$ .

Concept used:

To move the function up add outside the function as:

  $f\left(x\right)+b$ is $f\left(x\right)$ moved up by $b$ units.

Moving the function down works the same way as $f\left(x\right)-b$ is $f\left(x\right)$ move down $b$ units.

To shift the function left, add inside the function’s argument:

  $f\left(x+b\right)$ gives $f\left(x\right)$ shifted $b$ units to the left.

Shifting to the right work the same way as $f\left(x-b\right)$ is $f\left(x\right)$ shifted $b$ units to the rights.

A function $f\left(x\right)$ , a new function $g\left(x\right)=kf\left(x\right)$ , where $a$ is the constant, is a vertical stretch or vertical compression of the function $f\left(x\right)$ .

If $a>1$ then graph will be stretched.

If $0<a<1$ , then the graph will be compressed.

If $a<0$ ,then there will be combination of a vertical stretch or compression with a vertical reflection.

The graph can be sketch using graphing calculator like Desmos graphing calculator.

Calculation:

According to the given:

  $f\left(x\right)=x^5,g\left(x\right)={\left(3w\right)}^5-4$ .

Here the function $g\left(x\right)$ depend on the another variable $w$ which is not valid to be solved therefore, it should be replaced with $x$ as:

  $g\left(x\right)={\left(3x\right)}^5-4$ .

The transformation $f$ represent by the $g$ .

  $g\left(x\right)=kf\left(x\right)-b$ .

The graph of $g\left(x\right)$ is vertically stretched by the factor of $k$ and vertical shifted of $b$ unit down.

  $g\left(x\right)={\left(3x\right)}^5-4$ is the transformation graph of $f\left(x\right)=x^5$ , described by vertically stretched by the factor of $243$ and vertical shifted of $4$ unit down.

The graph can be sketch by vertically stretched by the factor of $k$ and vertical shifted of $b$ unit down as:

  

the graph of $g\left(x\right)$ is represented by green curve and graph of $f\left(x\right)$ is represented by the red curve.

The graph of the function $g\left(x\right)$ is vertically stretched by the factor of $243$ and vertical shifted of $4$ unit down.

Hence, the graph of the function $g\left(x\right)$ is vertically stretched by the factor of $243$ and vertical shifted of $4$ unit down.