Lesson 15
Features of Trigonometric Graphs (Part 1)
 Let’s compare graphs and equations of trigonometric functions.
15.1: Notice and Wonder: Musical Notes
Here are pictures of sound waves for two different musical notes:
What do you notice? What do you wonder?
15.2: Equations and Graphs
Match each equation with its graph. More than 1 equation can match the same graph.
Equations:
 \(y = \text\cos(\theta)\)
 \(y = 2\sin(\theta)3\)
 \(y = \cos\left(\theta + \frac{\pi}{2}\right)\)
 \(y = 3\sin(\theta)  2\)
 \(y = \sin(\theta\frac\pi 2)\)
 \(y = \sin(\theta+\pi)\)
 Find an equation for this graph using the sine function.
 Find another equation for the same graph using a cosine function.
15.3: Double the Sine
 Complete the table of values for the expression \(\sin(2\theta)\)
\(\theta\) 0 \(\frac{\pi}{12}\) \(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{2}\) \(\frac{3\pi}{4}\) \(\pi\) \(\frac{5\pi}{4}\) \(\frac{3\pi}{2}\) \(\frac{7\pi}{4}\) \(2\pi\) \(\sin(2\theta)\) 
Plot the values and sketch a graph of the equation \(y = \sin(2\theta)\). How does the graph of \(y = \sin(2\theta)\) compare to the graph of \(y = \sin(\theta)\)?

Predict what the graph of \(y = \cos(4\theta)\) will look like and make a sketch. Explain your reasoning.
Summary
We can find the amplitude and midline of a trigonometric function using the graph or from an equation. For example, let’s look at the function given by the equation \(y = 3\cos\left(\theta+\frac{\pi}{4}\right) + 2\). We can see that the midline of this function is 2 because of the vertical translation up by 2. This means the horizontal line \(y = 2\) goes through the middle of the graph. The amplitude of the function is 3. This means the maximum value it takes is 5, 3 more than the midline value, and the minimum value it takes is 1, 3 less than the midline value. The horizontal translation is \(\frac{\pi}{4}\) to the left, so instead of having, for example, a minimum at \(\pi\), the minimum is at \(\frac{3\pi}{4}\). Here is what the graph looks like:
Another type of transformation is one that affects the period and that is when a horizontal scale factor is used. For example, let's look at the equation \(y = \cos(2\theta)\) where the variable \(\theta\) is multiplied by a number. Here, 2 is the scale factor affecting \(\theta\). When \(\theta = 0\), we have \(2\theta = 0\) so the graph of this cosine equation starts at \((0,1)\), just like the graph of \(y = \cos(\theta)\). When \(x = \pi\), we have \(2\theta = 2\pi\) so the graph of \(y = \cos(2\theta)\) goes through two full periods in the same horizontal span it takes \(y = \cos(\theta)\) to complete one full period, as shown in their graphs.
Notice that the graph of \(y=\cos(2\theta)\) has the same general shape as the graph of \(y =\cos(\theta)\) (same midline and amplitude) but the waves are compressed together. And what if we wanted to give the graph of cosine a stretched appearance? Then we could use a horizontal scale factor between 0 and 1. For example, the graph of \(y=\cos(\frac{\theta}{6})\) has a period of \(12\pi\).
Glossary Entries
 amplitude
The maximum distance of the values of a periodic function above or below the midline.
 midline
The value halfway between the maximum and minimum values of a period function. Also the horizontal line whose \(y\)coordinate is that value.