Chapter 2.1, Problem 23E

To determine

To describe: The transformation of function $f(x)=x^2$ to $g(x)=\frac{1}{5}{x}^2-4$ and graph the functions.

Answer to Problem 23E

The function $g(x)=\frac{1}{5}{x}^2-4$ is reflected over $x-\text{axis}$ and horizontally shrunk by a factor of $\frac{1}{2}$ of the function $f(x)=x^2$ .

  

Explanation of Solution

Given information: Function transformed to $f(x)=x^2$ to $g(x)=\frac{1}{5}{x}^2-4$ .

Notice that the function is in the form of $g(x)=a{x}^2+k$ , where $a=\frac{1}{5}\text{and}k=-4$ .

Vertical shrink occurs when $0<a<1$ .

From the given function $g(x)=\frac{1}{5}{x}^2-4$ , we know that $a=\frac{1}{5}$ which is smaller than $1$ and greater than $0$ .

The constant $-4$ indicates the translation of $4$ units downwards.

Therefore, we can conclude that the function $g(x)=\frac{1}{5}{x}^2-4$ is a vertical shrink by a factor $\frac{1}{5}$ and translated $4$ units down.