1. 7-6 Amplitude, Midline, and Period of Waves

2. Before class begins, please download “Desmos Graphing Calculator” on your phone. (if you have a smart phone.)

3. The Graphs of Sine and Cosine
These are the graphs of 𝑦= sin 𝑥 and 𝑦= cos 𝑥 . We are going to study several things about them, but first we will answer, “Where do they come from?”

4. Imagine a flashlight shining a beam on the black spiral
Imagine a flashlight shining a beam on the black spiral. The red curve is what the shadow of spiral would look like on the wall. The blue curve is what it would look like on the floor. This spiral comes from moving the unit circle as the angle rotates.

5. Here is another visual of how the unit circle produces waves.

6. Midline and Amplitude
The midline of a wave is the “center of the wave” or the line that cuts the wave in half.
The amplitude is the highest or lowest point of the wave measured from the midline.

7. YOU WILL NEED TO DOWNLOAD DESMOS ONTO YOUR PHONE FOR THIS STEP
Midline Continued
The midline of a wave comes from adding a number to 𝑦= sin 𝑥 or 𝑦= cos 𝑥 .
YOU WILL NEED TO DOWNLOAD DESMOS ONTO YOUR PHONE FOR THIS STEP
Step 1: Type in sin (𝑥) +𝑘. You can find sin under the FUNCTIONS/FUNCS menu.
Step 2: Click the blue 𝑘 next to “add slider:”
Step 3: Adjust 𝑘 as you like. How does it affect the graph?
The line 𝑦=𝑘 is ALWAYS the midline of the wave.

8. YOU WILL NEED TO DOWNLOAD DESMOS ONTO YOUR PHONE FOR THIS STEP
Amplitude Continued
The amplitude of a wave comes from multiplying a number by 𝑦= sin 𝑥 or 𝑦= cos 𝑥 .
YOU WILL NEED TO DOWNLOAD DESMOS ONTO YOUR PHONE FOR THIS STEP
Step 1: Type in 𝑎sin (𝑥) . You can find sin under the FUNCTIONS/FUNCS menu.
Step 2: Click the blue 𝑎 next to “add slider:”
Step 3: Adjust 𝑎 as you like. How does it affect the graph?
|𝑎| is ALWAYS the amplitude of the wave.

9. Example 𝑦=2 cos 𝑥 +3 𝑦=5 sin 𝑥 −10 𝑦=7 cos 𝑥+3.2
Find the amplitude and midline of the following waves:
𝑦=2 cos 𝑥 +3
𝑦=5 sin 𝑥 −10
𝑦=7 cos 𝑥+3.2

10. Finding Amplitude and Midline from a Graph
What are the amplitude and midline for the graph at right?
To find the midline of the wave, we average the highest 𝑦-value and the lowest 𝑦-value together (because the average is the “middle”). The highest 𝑦-value is −2 and the lowest 𝑦-value is −6, so their average is −2+(−6) 2 =−4. The midline is the line 𝑦=−4.
The amplitude is how far away the highest 𝑦-value is from the midline. The highest 𝑦-value is −2 which is 2 away from the midline. So the amplitude is 2.

11. Example
Find the amplitude and midline of the wave.

12. The Period of a Wave
The period of a wave is how long it takes for the wave to repeat itself.
Using your unit circle, determine the period of sin 𝑥 and cos 𝑥 . (When do they start repeating?)

13. Changing the Period (Desmos)
The period is changed by multiplying 𝑥 by a number.
Step 1: Type in sin (𝑏𝑥) . You can find sin under the FUNCTIONS/FUNCS menu.
Step 2: Click the blue 𝑏 next to “add slider:”
Step 3: Adjust 𝑏 as you like. How does it affect the graph?
𝑏 affects the period, but the period is ALWAYS 2𝜋 𝑏 .

14. Finding the Period from an Equation
What is the period of 𝑦=3 cos 4𝑥 ?
The period 𝑇 is 2𝜋 𝑏 . 𝑇= 2𝜋 4 = 𝜋 2 so the wave repeats itself every 𝜋 2 units. Below is its graph.

15. Examples 𝑦=−1.5 cos 3𝑥 −1 𝑦=6 sin 0.5𝑥 +9
Find the amplitude, midline, and period of the following:
𝑦=−1.5 cos 3𝑥 −1
𝑦=6 sin 0.5𝑥 +9

16. Finding the Period from a Graph
Finding the period from a graph can be difficult because it is often a multiple of 𝜋, but if the graph is properly marked, it is easy to do. Below is the same wave with different markings. Let’s find the period for both.

17. Examples
Find the period of the following waves.

18. Is it a sine wave or a cosine wave?
On Desmos, type in sin⁡(𝑥) for your first function and cos⁡(𝑥) for your second function. How are they similar? How are they different?
𝑦= sin 𝑥 and 𝑦= cos 𝑥 are the same wave, they are just shifted from each other!
To decide whether to use sin 𝑥 or cos 𝑥 , we just need to look at the 𝑦-axis. If the wave crosses the midline at the 𝑦-axis, it is sin 𝑥 . If the wave has its highest or lowest point at the 𝑦-axis, it is cos 𝑥 .

19. Example Is it sine or cosine? Find the midline. Find the amplitude.
Find the period.

20. 𝑦=𝑎∙ sin 𝑏𝑥 +𝑘 𝑦=𝑎∙ cos 𝑏𝑥 + 𝑘
Writing Equations
The general form of sine and cosine functions is as follows.
𝑦=𝑎∙ sin 𝑏𝑥 +𝑘 𝑦=𝑎∙ cos 𝑏𝑥 + 𝑘
Where 𝑎 is the amplitude 𝑘 is the midline value and 2𝜋 𝑏 is the period.
Write an equation for the previous example.
Write an equation for a sine wave with amplitude 10, midline 𝑦=−6 and period 2𝜋 5 .