Chapter 2.5, Problem 4AYU

To determine

Whether the graph of $y=-f\left(x\right)$ is obtained by areflection about thex- axis of the graph of $y=f\left(x\right)$.

Answer to Problem 4AYU

The statement is true.

Explanation of Solution

Consider $y=f\left(x\right)$

When the graph is reflected about x- axis, then the transformation is $\left(x,y\right)\Rightarrow \left(x,-y\right)$

So $h\left(x\right)=-f\left(x\right)$

When the graph is reflected abouty- axis, then the transformation is $\left(x,y\right)\Rightarrow \left(-x,y\right)$

So $h\left(x\right)=f\left(-x\right)$

Now, consider the graph of $y=-f\left(x\right)$ .

Here, the graph is of the form $h\left(x\right)=-f\left(x\right)$ ,

The graph of $y=-f\left(x\right)$ is obtained by areflection about thex- axis of the graph of $y=f\left(x\right)$ .