Chapter 2.1, Problem 22E

To determine

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

Answer to Problem 22E

The function $g(x)=-{(2x)}^2$ 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 $g(x)=-{(2x)}^2$ .

Notice that the function is in the form of $g(x)={(ax)}^2$ , where $a=2$ .

The function $g(x)=-{(2x)}^2$ flips over the $x-\text{axis}$ because of the negative sign.

Horizontal shrink occurs when $a>1$ .

Therefore, we can conclude that the function $g(x)=-{(2x)}^2$ is reflected over $x-\text{axis}$ and shrinks horizontally by the factor $\frac{1}{2}$ .