Chapter 9.3, Problem 3BCYP

To determine

Describe how the graph of each function is related to the graph of $f(x)=x^2\text{and}g(x)=-\frac{1}{5}{x}^2+3$

Answer to Problem 3BCYP

The graph of $g(x)=-\frac{1}{5}{x}^2+3$ is the graph of $f(x)=x^2$ reflected across the x − axis, vertically compressed by a factor of $\frac{1}{5}$ and translated up 3 units

Explanation of Solution

Given: The functions $f(x)=x^2\text{and}g(x)=-\frac{1}{5}{x}^2+3$

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 function can be written $f(x)=a{x}^2+c$ , where   $a=-\frac{1}{5}$ and c = +3, the graph is reflected across the x − axis and vertically compressed by a factor $\frac{1}{5}$. Since c = +3, the graph is translated up 3 units. So, the graph of $g(x)=-\frac{1}{5}{x}^2+3$ is the graph of $f(x)=x^2$ reflected across the x − axis, vertically compressed by a factor of $\frac{1}{5}$ and translated up 3 units.

Thus, the graph of $g(x)=-\frac{1}{5}{x}^2+3$ is the graph of $f(x)=x^2$ reflected across the x − axis, vertically compressed by a factor of $\frac{1}{5}$ and translated up 3 units