Q. 108 - Precalculus Enhanced with Graphing Utilities

Question

Open the Period applet. On the screen you will see a slider. Move the point along the slider to see the role $\omega$ plays in the graph of $f\left(x\right)=\sin \left(\omega x\right)$. Pay particular attention to the key points matched by color on each graph. For convenience the graph of $g\left(x\right)=\sin x$ is shown as a dashed, gray curve 

Step-by-Step Solution

Answer

The graph of $g\left(x\right)=\sin x$ is horizontally compressed by a factor of $\frac{1}{\omega }$ to get the graph of $f\left(x\right)=\sin \left(\omega x\right)$.

Step 1. Use the applet

The slider is moved and one of the result can be seen as

Step 2. Observation

It can be seen that the new graph is compressed.

Its cycle length is decreased.

So it can be seen that the graph of $g\left(x\right)=\sin x$ is horizontally compressed by a factor of $\frac{1}{\omega }$ to get the graph of $f\left(x\right)=\sin \left(\omega x\right)$..