Chapter 8.3, Problem 11Q

To determine

To describe: The transformation from the graph of the function $f\left(x\right)=\frac{1}{2}{x}^2-2$ to the graph of $g\left(x\right)=f\left(x\right)-6$ . Also sketch the graph of the functions and express the function g in terms of x.

Answer to Problem 11Q

The graph of the function $g\left(x\right)=f\left(x\right)-6$ is the vertical translation 6 units down of the graph of $f\left(x\right)=\frac{1}{2}{x}^2-2$ . The function g is expressed as, $g\left(x\right)=\frac{1}{2}{x}^2-8$ . The red curve represents the graph of $f\left(x\right)=\frac{1}{2}{x}^2-2$ and blue curve represents the graph of $g\left(x\right)=f\left(x\right)-6$ .

  

Explanation of Solution

Given information:

The functions $f\left(x\right)=\frac{1}{2}{x}^2-2$ and $g\left(x\right)=f\left(x\right)-6$ .

Formula used:

To graph the function $f\left(x\right)=a{x}^2+k$ where $k>0$ , it is the vertical translation k units up of the graph $f\left(x\right)=a{x}^2$ .

To graph the function $f\left(x\right)=a{x}^2+k$ where $k<0$ , it is the vertical translation $\left|k\right|$ units down of the graph $f\left(x\right)=a{x}^2$ .

Calculation:

Consider the functions $f\left(x\right)=\frac{1}{2}{x}^2-2$ and $g\left(x\right)=f\left(x\right)-6$ .

Recall that to graph the function $f\left(x\right)=a{x}^2+k$ where $k<0$ , it is the vertical translation $\left|k\right|$ units down of the graph $f\left(x\right)=a{x}^2$ .

The graph of the function $g\left(x\right)=f\left(x\right)-6$ is the vertical translation 6 units downward of the graph of $f\left(x\right)=\frac{1}{2}{x}^2-2$ .

The graph of the quadratic functions $f\left(x\right)=\frac{1}{2}{x}^2-2$ and $g\left(x\right)=f\left(x\right)-6$ is sketched with help of ordered pairs.

Construct a table of values with values of independent and dependent variable.

For some values of x evaluate the corresponding values of the function.

  $\begin{array}{cccccc}x& -2& -1& 0& 1& 2\\ f\left(x\right)=\frac{1}{2}{x}^2-2& 0& -1.5& -2& -1.5& 0\\ g\left(x\right)=f\left(x\right)-6& -6& -7.5& -8& -7.5& -6\end{array}$

Plot the ordered pairs on a coordinate plane and join the points to obtain the smooth curve.

  

The red curve represents the graph of $f\left(x\right)=\frac{1}{2}{x}^2-2$ and blue curve represents the graph of $g\left(x\right)=f\left(x\right)-6$ .

Now, take the function $g\left(x\right)=f\left(x\right)-6$ , substitute $f\left(x\right)=\frac{1}{2}{x}^2-2$ in the function and simplify,

  $\begin{array}{c}g\left(x\right)=f\left(x\right)-6\\ =\frac{1}{2}{x}^2-2-6\\ =\frac{1}{2}{x}^2-8\end{array}$

Thus, the graph of the function $g\left(x\right)=f\left(x\right)-6$ is the vertical translation 6 units downward of the graph of $f\left(x\right)=\frac{1}{2}{x}^2-2$ . The function g is expressed as, $g\left(x\right)=\frac{1}{2}{x}^2-8$ .