Chapter 4.7, Problem 17E

To determine

To find: the rule for the function $g$ and graph the function $g$ with description of graph as a transformation of the graph $f$ function.

Answer to Problem 17E

The graph of the function $g\left(x\right)$ shifted $2$ units left.

Explanation of Solution

Given:

  $f\left(x\right)=x^4+1,g\left(x\right)=f\left(x+2\right)={\left(x+2\right)}^4+1$ .

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 using graphing calculator like Desmos graphing calculator.

Calculation:

According to the given:

  $f\left(x\right)=x^4+1,g\left(x\right)=f\left(x+2\right)={\left(x+2\right)}^4+1$ .

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

  $g\left(x\right)=f\left(x+a\right)$ .

The graph can be sketch as $f\left(x\right)$ shifted $a$ units left.

  $g\left(x\right)={\left(x+2\right)}^4+1$ is the transformation graph of $f\left(x\right)=x^4+1$ , the function $g\left(x\right)$ is horizontal shifted $2$ units left.

The graph can be sketch by horizontal shifting $2$ unit to the left.

  

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

The graph of the function $g\left(x\right)$ shifted $2$ units left.

Hence, the graph of the function $g\left(x\right)$ shifted $2$ units left.