Q. 64 - Precalculus Enhanced with Graphing Utilities

Question

Exploration Graph $y=\sqrt{x}$. Then on the same screen graph $y=\sqrt{-x}$. What pattern do you observe? Now try $y=2x+1$ and $y=2(-x)+1$. What do you conclude? 

Step-by-Step Solution

Answer

The graph of $y=\sqrt{-x}$ is the reflection about the  $y$-axis of the graph of $y=\sqrt{x}$.

The same can be seen for the graph of $y=2x+1$ and  $y=2(-x)+1$.

It can be concluded that the graph of $y=f(-x)$ is the reflection about the $y$ 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 $y=\sqrt{x}$ and $y=\sqrt{-x}$. 

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

So it can be said that the graph of $y=\sqrt{-x}$ is the reflection about the $y$ axis of the graph of $y=\sqrt{x}$.

Step 2. Graph the second set of functions

In the same coordinate plane draw the graphs of $y=2x+1$ and $y=2(-x)+1$. 

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

So it can be said that the graph of $y=2(-x)+1$ is the reflection about the $y$ axis of the graph of $y=2x+1$.

Step 3. Conclusion

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