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:

graph of 2 curves. curve a has greater amplitude, curve a has a shorter period, both curves have midline on he x axis. 

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.


  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. \(y = \sin(\theta+\pi)\)


graph of sine function. midline at y = -3. amplitude = 2.


graph of sine function with midline of y = -2, amplitude = 3. 


graph of cosine function shifted by the fraction pi over 2 to the left. graph could also be the sine function shifted pi to the left. 


graph of sine function shifted by the fraction pi over 2 to the right. graph could also be the cosine function reflected over the horizontal axis.


  1. Find an equation for this graph using the sine function.
  2. Find another equation for the same graph using a cosine function.
Function on a coordinate grid, origin O.

15.3: Double the Sine

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

    horizontal axis, theta, scale 0 to 2 pi, by the fraction pi over 4. vertical axis, y, scale -1 to 1, by 1's. 
  3. Predict what the graph of \(y = \cos(4\theta)\) will look like and make a sketch. Explain your reasoning.

    horizontal axis, theta, scale 0 to 2 pi, by the fraction pi over 4. vertical axis, y, scale -1 to 1, by 1's. 


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:

graph of cosine function. midline at y = 2. amplitude = 3. horizontal translation of the fraction pi over 4 to the left. 

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.

graph of y = cosine of 2 theta. period= pi, amplitude - 1. graph of y = cosine of theta. period = 2 pi, amplitude = 1. 

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.