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## Integrated math 3

### Course: Integrated math 3>Unit 8

Lesson 9: Amplitude, midline, & period

# Features of sinusoidal functions

Sal introduces the main features of sinusoidal functions: midline, amplitude, & period. He shows how these can be found from a sinusoidal function's graph. Created by Sal Khan.

## Want to join the conversation?

• Hello! How do I determine if a function has a period algebraically? Thanks! •   There is a way to do this, but to be honest it is much easier to do graphically. Also, the math involved can get fairly advanced and rather hard to avoid making errors with.

But here is how you would do it:
The function f(x) is periodic if and only if:
f(x+nL) - f(x) = 0, where n is any integer and L is some constant other than 0.
If the only solution for L is 0, then the function is NOT periodic.

Thus, set n=1 and solve for L.
After doing so, demonstrate that
f(x+nL) - f(x) = 0, for integer values of n.

So, that is how you would determine this mathematically. I don't recommend attempting it because it is quite difficult and often involves nonreal complex exponents or complex logarithms.

Note: there are some functions that have more than one period, but these are really advanced level math and you probably won't encounter them at this level of study.
• How do I know whether I must use midline = (max val + min val) / 2 or (max val - min val) / 2? I'm really confused •  Instead of relying on formulas that are so alike that they're confusing (to me, too!), maybe try to think it through each time (at least in the beginning) until it gets more familiar). I had a LOT of difficulty with this type of problem and I found that I had to go slowly and think things through each step EVERY time I did a problem.

Here's a method I found helpful. Maybe it will be of use to you.
I know that the midline lies halfway between the max and the min. So I need to get the total height (by subtracting the min from the max). (That gives me ( 4 - (-2) ). Now I divide by 2. (That gives me 3.) By definition that is the AMPLITUDE. Now I can either add that to the min (or subtract it from the max), and where I end up is the MIDLINE ( at 1 ).
Good luck!
• Can someone please explain how to find the midline of a sinusoidal function from its equation, instead of the graph? • y = A sin (B(x - C)) + D is a general format for a sinusoidal function. The number in the D spot represents the midline. The equation of the midline is always 'y = D'.

Example:
y = 3 sin(2(x - π)) - 5 has a midline at y = -5
• What are sinusoidal functions? • A sinusoidal function is one with a smooth, repetitive oscillation. "Sinusoidal" comes from "sine", because the sine function is a smooth, repetitive oscillation. Examples of everyday things which can be represented by sinusoidal functions are a swinging pendulum, a bouncing spring, or a vibrating guitar string.

I hope this helps. Good luck!
• Do you have any videos that actually talk about the graphs of trig functions? If so please post as soon as possible. This title is very misleading. I assumed you would teach what the trig functions looked like but it seemed more like you expected us to know it already :(. On the next video I was so frustrated because I did not even know what -1/2 cos(3x) meant. I didn't even know these things could be graphed. I thought you only used for triangles or something. What is all this graphing stuff? SO frustrated :/ • If you watch the videos in the preceding section headed "Unit circle definition of trig functions", you will appreciate that the cosine and sine functions take an angle as the input value, and give output values that repeat every so often, and that always remain within the values -1 and 1. If, instead of thinking about the x and y coordinates of points on the unit circle, you decide to plot a graph with angle on the x-axis, with the y axis being the cosine or sine of the variable x, you will obtain a pattern like the one in this video.

So, this is the video where Sal is showing you what the trig functions look like. If the maximum value of the cosine or sine of any angle is 1, and the minimum value is -1, then the amplitude of these functions is 1, and any function that is a multiple of one of these functions will have an amplitude of 1 times that multiple, or -1/2 in the case of cos(3x).

Edit: Actually, all this is made more explicit in this video: https://www.khanacademy.org/math/trigonometry/trig-function-graphs/trig_graphs_tutorial/v/we-graph-domain-and-range-of-sine-function
• Hello, I'm just wondering why Sal choice to use the Midline to find the period: is this always the case? or is it just easier to use the Midlines y value instead?

Thank you! • Hi Daniel,
No, you do not have to use the midline to find the period. You can find the period by going from peak to peak, or trough to trough, or midline to midline. If you use midline of course you will need to keep in mind that you will need to skip a midline (because the midlines you measure from must be going the same direction).

Hope this helps,
- Convenient Colleague
• Can the "midline" also be called the "sinusoidal axis"? • I have watched this video over and over and i get amplitude and midline but finding the period makes no sense to me. is there a formula i can use? • Periods of a sinusoidal functions are very very confusing so I can empathize with you on that.

Let's just say the given is from the midline to maximum, with a distance of 3.

Whenever you are given a mid-line to a maximum/minimum, always multiply that distance by 4. A graphic in the practice problems explains why.

3*4 = 12. Now for every time you want to find the period, use this formula.

2pi / (that number you multipled by 4).
//always use this formula when finding the period !

So now you have 2pi/12. Simplifying that, you get pi/6. boom, period !

Also if you have given like a maxiumum to maximum or minimum to minimum, instead of multiplying by 4, multiply by 2. Again the graphic shows a visual interpretation.

Hopefully that helps ! This is how I interpreted it as.  