Chapter 9.3, Problem 5CYU

To determine

Describe how the graph of each function is related to the graph of $f(x)=x^2\text{and}g(x)=-4x^2$

Answer to Problem 5CYU

The graph of $y=-4x^2$ is the graph of $y=x^2$ reflected across the x − axis and vertically stretched by a factor 4.

Explanation of Solution

Given: The functions $f(x)=x^2\text{and}g(x)=-4x^2$

Concept Used:

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

The graph of $f(x)=a{x}^2$ stretches or compress the parent graph vertically The coefficient of $x^2$ term is negative, so the graph is reflected across the x − axis. Since $a=4$ , the graph is vertically stretched. So, the graph of $y=-4x^2$ is the graph of $y=x^2$ reflected across the x − axis and vertically stretched by a factor 4.

Thus, the graph of $y=-4x^2$ is the graph of $y=x^2$ reflected across the x − axis and vertically stretched by a factor 4.