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?

Short answer

The graph of $y=-x^2$is the reflection about the $x$-axis of the graph of role="math" localid="1645847739257" $y=x^2$.

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-by-step solution

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