Q. 63 - Precalculus Enhanced with Graphing Utilities

Question

Exploration Graph $y=x^2$. Then on the same screen graph $y=-x^2$. What pattern do you observe? Now try $y=\left|x\right|$ and $y=-\left|x\right|$. What do you conclude? 

Step-by-Step Solution

Answer

The graph of $y=-x^2$ is the reflection about the  $x$-axis of the graph of $y=x^2$.

The same can be seen for the graph of $y=\left|x\right|$ and  $y=-\left|x\right|$.

It can be concluded that the graph of $y=-f\left(x\right)$ is the reflection about the $x$ axis of the graph of $y=f\left(x\right)$.

Step 1. Graph the first set of functions

In the same coordinate plane draw the graphs of $y=x^2$ and $y=-x^2$.

It can be seen that both the graphs are reflections of each other about the $x$ axis.

So it can be said that the graph of $y=-x^2$ is the reflection about the $x$ axis of the graph of $y=x^2$.

Step 2. Graph the second set of functions

In the same coordinate plane draw the graphs of $y=\left|x\right|$ and $y=-\left|x\right|$.

Again, it can be seen that both the graphs are reflections of each other about the $x$ axis.

So it can be said that the graph of $y=-\left|x\right|$ is the reflection about the $x$ axis of the graph of $y=\left|x\right|$.

Step 3. Conclusion

From the observation of the above two sets of functions, we can come to a conclusion that the graph of $y=-f\left(x\right)$ is the reflection about the $x$ axis of the graph of $y=f\left(x\right)$.