# 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:

1. $$y = \text-\cos(\theta)$$
2. $$y = 2\sin(\theta)-3$$
3. $$y = \cos\left(\theta + \frac{\pi}{2}\right)$$
4. $$y = 3\sin(\theta) - 2$$
5. $$y = \sin(\theta-\frac\pi 2)$$
6. $$y = \sin(\theta+\pi)$$

1. Find an equation for this graph using the sine function.
2. Find another equation for the same graph using a cosine function.

### 15.3: Double the Sine

1. Complete the table of values for the expression $$\sin(2\theta)$$
 $$\theta$$ $$\sin(2\theta)$$ 0 $$\frac{\pi}{12}$$ $$\frac{\pi}{6}$$ $$\frac{\pi}{4}$$ $$\frac{\pi}{2}$$ $$\frac{3\pi}{4}$$ $$\pi$$ $$\frac{5\pi}{4}$$ $$\frac{3\pi}{2}$$ $$\frac{7\pi}{4}$$ $$2\pi$$
2. 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 $$y = \sin(\theta)$$?

3. 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.