Chapter 3.7, Problem 3E

To determine

To write: Three types of transformations of modulus functions with description.

Explanation of Solution

Concept used:

The function $y=f\left(x+c\right)$ moves left by $c$ unit when $c>0$ and moves right by $c$ unit when $c<0$ with base function $f\left(x\right)$ . This is horizontal shifting in the function.

The function $y=f\left(x\right)+c$ moves upward by $c$ unit when $c>0$ and moves downward by $c$ unit when $c<0$ with base function $f\left(x\right)$ . This is vertical shifting in the function.

The function $y=cf\left(x\right)$ stretches in the $y$ direction when $c>1$ and compresses when $0<c<1$ with base function $f\left(x\right)$ .

The function reflect itself about $x$ axis if the base function is $f\left(x\right)=\left|x\right|$ then, the reflected function is $-f\left(x\right)$ .

The absolute value function can be written as follows:

  $\left|x\right|=\{\begin{array}{l}x\text{ if }x\ge 0\\ -x\text{ if }x<0\end{array}$

Use first concept for the horizontal shifting in the modulus shifting where $f\left(x\right)=\left|x\right|$ .

Choose the value of $c=1$ .

The transformation for $c=1$ in the horizontal shifting is as follows:

  $\begin{array}{c}f\left(x+c\right)=\left|x+c\right|\\ =\left|x+1\right|\end{array}$

The transformation indicates the function moves left by 1 unit.

Now use second concept for the vertical shifting in the modulus shifting where $f\left(x\right)=\left|x\right|$ .

Choose the value of $c=1$ .

The transformation for $c=1$ in the vertical shifting is as follows:

  $\begin{array}{c}f\left(x\right)+c=\left|x\right|+c\\ =\left|x\right|+1\end{array}$

The transformation indicates the function moves upward by 1 unit.

Now use third concept for the vertical shifting in the modulus shifting where $f\left(x\right)=\left|x\right|$ .

Choose the value of $c=2$ .

The transformation of $c=2$ for third concept is as follows:

  $\begin{array}{c}f\left(cx\right)=c\left|x\right|\\ =2\left|x\right|\end{array}$

The transformation indicates the function stretches itself by two times.