Chapter 3.5, Problem 49PPS

To determine

To find: The translation of given $f\left(x\right)$

Answer to Problem 49PPS

The graph of $a\cdot f\left(x\right)$ for $a=-1$ is reflection of graph of $f\left(x\right)$ about x-axis.

Explanation of Solution

Given information:

  $f\left(x\right)$

  $a\cdot f\left(x\right)$ for $a=-1$

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If the function $y=f\left(x\right)$ is translated to $y=-f\left(x\right)$ , then the graph of $f\left(x\right)$ is reflected about x-axis.

From the given information,

Let $g\left(x\right)$ be defined such that $g\left(x\right)=a\cdot f\left(x\right)$

Since $a=-1$

  $\Rightarrow g\left(x\right)=-f\left(x\right)$ , therefore, the graph of $f\left(x\right)$ is reflected about x-axis.

Therefore, the graph of $g\left(x\right)$ is same as the graph of $f\left(x\right)$ which is mirror image about x-axis.