Chapter 8.2, Problem 14E

To determine

To describe:Describe the transformation from the graph of $f$ to the graph of $g$ . Then graph $f$ and $g$ in the same coordinate plane. Write an equation that represents $g$ in terms of $x$.

Answer to Problem 14E

The graph of $g$ is a vertical shift 4 units down of the graph of $f$.

Graph:

  

The equation $g(x)=\frac{1}{2}{x}^2-3$ represents $g$ in terms of $x$.

Explanation of Solution

Giveninformation:

The function are $f(x)=\frac{1}{2}{x}^2+1$ and $g(x)=f(x)-4$.

Graph:

Given that $g(x)=f(x)-4$ . So, the graph of $g$ is a vertical shift 4 units down of the graph of $f$.

Now, to graph the functions, let’s make the table values as follows:

$x$	$-2$	$-1$	0	1	2
$f(x)$	3	1.5	1	1.5	3
$g(x)$	$-1$	$-2.5$	$-3$	$-2.5$	$-1$

Now, plotting the ordered pairs and drawing a smooth curve through the points, we get the following graph.

  

Now,

  $\begin{array}{l}g(x)=f(x)-4\\ \Rightarrow g(x)=\frac{1}{2}{x}^2+1-4\\ \Rightarrow g(x)=\frac{1}{2}{x}^2-3\end{array}$

So, the equation $g(x)=\frac{1}{2}{x}^2-3$ represents $g$ in terms of $x$.