Chapter 4.7, Problem 13E

To determine

To find: the transformation of $f$ represented by $g$ and graph the function.

Answer to Problem 13E

The graph of the function $g\left(x\right)$ vertically stretched with the factor of $5$ and $1$ unit up from the graph of $f\left(x\right)$ .

Explanation of Solution

Given:

  $f\left(x\right)=x^3,g\left(x\right)=5x^3+1$ .

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.

Or

  $g\left(x\right)=f\left(kx\right)$

The graph can be sketch by horizontally shrinking, $f\left(x\right)$ by the factor of $\frac{1}{k}$ .

Calculation:

According to the given:

  $f\left(x\right)=x^3,g\left(x\right)=5x^3+1$ .

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

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

The graph will be vertical stretched, $g\left(x\right)$ by the factor of $k$ and vertical shifted by $b$ unit up.

  $g\left(x\right)=5x^3+1$ is the transformation graph of $f\left(x\right)=x^3$ , described by a vertical stretched by a factor of $5$ and vertical shifted by $1$ unit up.

The graph can be sketch by vertical stretched, $g\left(x\right)$ by the factor of $k$ 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 $g\left(x\right)$ vertically stretched with the factor of $5$ and $1$ unit up from the graph of $f\left(x\right)$ .

Hence, graph of the function $g\left(x\right)$ vertically stretched with the factor of $5$ and $1$ unit up from the graph of $f\left(x\right)$ .