Chapter 2.1, Problem 24E

To determine

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

Answer to Problem 24E

The function $g(x)=\frac{1}{2}{(x-1)}^2$ 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: The graph of the function $f(x)=x^2$ transformed to $g(x)=\frac{1}{2}{(x-1)}^2$ .

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

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

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

  ${(x-1)}^2$ Shows the translation of $1$ unit right.

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