Chapter SH, Problem 9.3.7EP

To determine

The relation between the graph of given function and graph of $f\left(x\right)=x^2.$

Answer to Problem 9.3.7EP

The graph of $g\left(x\right)=\frac{3}{4}{x}^2+4$ is compressed vertically and then translated 4 units up.

Explanation of Solution

Given:

The given function is:

  $g\left(x\right)=\frac{3}{4}{x}^2+4$

Calculation:

Consider the function $g\left(x\right)=\frac{3}{4}{x}^2+4$ :

In the graph of $g\left(x\right)=x^2+c$ , $x^2$ is translated $\left|c\right|$ units up.

In the graph of $g\left(x\right)=x^2-c$ , $x^2$ is translated $\left|c\right|$ units down.

The value of c in $g\left(x\right)=\frac{3}{4}{x}^2+4$ is 4.

The graph of $g\left(x\right)=\frac{3}{4}{x}^2+4$ is translated 4 units up.

Consider the function $g\left(x\right)=a{x}^2$ :

For this function, the graph is stretched vertically if $\left|a\right|>1$ and is compressed vertically if $0<\left|a\right|<1.$

In the given function,

  $\begin{array}{l}g\left(x\right)=\frac{3}{4}{x}^2+4\\ a=\frac{3}{4}\end{array}$

So, the graph is compressed vertically as, $0<\left|a\right|<1.$

Therefore, the graph of $g\left(x\right)=\frac{3}{4}{x}^2+4$ is compressed vertically and then translated 4 units up.

The graph will be: