Chapter 9.3, Problem 6CYU

To determine

Describe how the graph of each function is related to the graph of $f(x)=x^2\text{and}h(x)=-x^2-2$

Answer to Problem 6CYU

The graph of $y=-x^2-2$ is the graph of $y=x^2$ reflected across the x − axis and translated down 2 units.

Explanation of Solution

Given: The functions $f(x)=x^2\text{and}h(x)=-x^2-2$

Concept Used:

  $\begin{array}{l}\text{Vertical}\text{Stretch}\text{and}\text{Vertical}\text{Compression:}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{a}\text{>1}\text{,}\text{Srtetch}\text{the}\text{graph}\text{pf}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\text{.}\\ \text{y}\text{=}\text{a}\text{f}\text{(x)}\text{,}\text{0}\text{<}\text{a}\text{<1}\text{,}\text{Compress}\text{the}\text{graph}\text{pf}\text{f(x)}\text{vertically}\text{by}\text{a}\text{factor}\text{of}\text{a}\end{array}$

The functions can be written as $f(x)=a{x}^2+c$ , where $a=-1\text{and}c=-2$ Since $a=-1$ , the coefficient of $x^2$ is negative, the graph is reflected across the x − axis. Since $c=-2$ , the graph is translated down 2 units. So, the graph of $y=-x^2-2$ is the graph of $y=x^2$ reflected across the x − axis and translated down 2 units.

Thus, the graph of $y=-x^2-2$ is the graph of $y=x^2$ reflected across the x − axis and translated down 2 units.