Chapter 3.7, Problem 15E

To determine

To graph: The function $k(x)=-3\left|x-6\right|$ and also compare the graph $k(x)=-3\left|x-6\right|$ with the graph $f(x)=\left|x-6\right|$ .

Explanation of Solution

Given information:

Two functions are $k(x)=-3\left|x-6\right|$ and $f(x)=\left|x-6\right|$ .

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 $f\left(x\right)=\left|x\right|$ .

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

The function reflect itself about $x$

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

Graph:

The given function is $k(x)=-3\left|x-6\right|$ .

There is requirement is to find the coordinates of the function.

Substitute 0 for $x$ in the function $k(x)=-3\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}k(x)=-3\left|0-6\right|\\ =-3\cdot 6\\ =-18\end{array}$

Substitute 2 for $x$ in the function $k(x)=-3\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}k(x)=-3\left|2-6\right|\\ =-3\cdot 4\\ =-12\end{array}$

Substitute 6 for $x$ in the function $k(x)=-3\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}k(x)=-3\left|6-6\right|\\ =-3\cdot 0\\ =0\end{array}$

Substitute 8 for $x$ in the function $k(x)=-3\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}k(x)=-3\left|8-6\right|\\ =-3\cdot 2\\ =-6\end{array}$

The coordinates are obtained as $\left(6,0\right),\left(2,-12\right)\text{,}\left(8,-6\right)\text{ and }\left(0,-18\right)$ .

Draw the graph of function $k(x)=-3\left|x-6\right|$ as shown in Figure 1.

  

Now draw the graph of function $f(x)=\left|x-6\right|$ for comparison.

Substitute 2 for $x$ in the function $f(x)=\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}f(x)=\left|2-6\right|\\ =\left|-4\right|\\ =4\end{array}$

Substitute 0 for $x$ in the function $f(x)=\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}f(x)=\left|0-6\right|\\ =\left|-6\right|\\ =6\end{array}$

Substitute 6 for $x$ in the function $f(x)=\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}f(x)=\left|6-6\right|\\ =0\end{array}$

Substitute 8 for $x$ in the function $f(x)=\left|x-6\right|$ to obtain the value of $y$ .

  $\begin{array}{c}f(x)=\left|8-6\right|\\ =2\end{array}$

The coordinates are obtained as $\left(2,4\right)\text{,}\left(0,6\right),\left(6,0\right)\text{ and }\left(8,2\right)$ .

Draw the graph of the function $f(x)=\left|x-6\right|$ as shown in Figure 2.

  

Compare the graph of the equation $k(x)=-3\left|x-6\right|$ with the function $f(x)=\left|x-6\right|$ by the use the transformation.

It can be seen that $\left|x-6\right|$ is common in both the functions.

Take the base function as $\left|x-6\right|$ .

It can be observed from the Figure 1 and Figure 2 $k(x)=-3\left|x-6\right|$ stretches in the direction of y-axis as $\left|x-6\right|$ is multiplied $3$ .

The function $k(x)=-3\left|x-6\right|$ is multiplied by $-3$ to $\left|x-6\right|$ as a result the function will be reflected below the x -axis.

At last the graph of function $f(x)=\left|x-6\right|$ is placed above the x -axis and in the positive direction of y-axis and on the other hand, the graph of the function $k(x)=-3\left|x-6\right|$ is placed below the x -axis and compressed in the direction of negative y-axis.