Finding the Sinusoidal Function

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  • 0:01 A Sound Wave
  • 1:06 Amplitude
  • 1:56 Period
  • 3:28 Shifts
  • 4:52 Our Sinusoidal Function
  • 6:23 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will be able to write a sinusoidal function to represent any wave that you see. Learn what parts of the function make the wave shift up, down, to the left, and to the right.

A Sound Wave

What did you just hear? Pay attention and you will hear all kinds of sounds. Do you like listening to music? Music, just like all the other sounds you hear, is sent to your ears via a sound wave. In this video lesson, we are going to learn how to take such a wave and turn it into a sinusoidal function, a function using the sine function.

Look at this graph:

Sinusoidal Function
sinusoidal function

This is the graph of the sinusoidal function y = sin (x). This is also an example of how a sound wave looks. Different sounds will look different. A high-pitched sound for example, might look like this:

Graph of High-Pitched Sound
sinusoidal function

This is a graph of the function y = sin (10x). This wave goes up and down much faster than the first wave we saw. We can note any change in our sound wave by adding or changing numbers in our sinusoidal function. Do you want to see how?


Okay! Let's begin by talking about the amplitude or height of our sinusoidal function. If we see a sound wave that goes up to 4 and goes to -4, we say that it has an amplitude of 4. To show this in our sinusoidal function, we write the number 4, the amplitude, in front of our sine function. So, a sinusoidal function of y = 4 sin (x) will have an amplitude of 4. The function y = sin (x) has an amplitude of 1 since if no number is shown, there is always a 1 there. The graph of y = 4 sin (x) looks like this:

y = 4 sin (x)
sinusoidal function

Do you see how the height of this wave goes up to 4 and then goes down to -4? Our amplitude tells us how high the wave goes from its midpoint. In math, we label the amplitude with a capital A.


Next, comes the period or length of one cycle before a sinusoidal function repeats itself. I showed you a bit of this a little bit at the beginning when we talked about a sound wave with higher pitch. You might have noticed that we multiplied our variable with a number that changed the period of the sound wave. How does this number change the period? The normal period of a sine wave is 2pi. When we multiply our variable with a number, our period now becomes our normal period divided by that number. If we label this number with the letter B, we can now say that the period changes to 2pi/B. So, if we multiply our variable by 10, our period changes to 2pi/10 = pi/5.

y = sin (10x)
sinusoidal function

This is the graph of y = sin (10x). Notice the shorter period. What do you think will happen when we divide our variable by a number? Let's see what happens when we divide our variable by 2.

y = sin (x/2)
sinusoidal function

Yes, our period increases. Let's do the math. Our period changes to 2pi/B = 2pi/(1/2) = 2pi*2 = 4pi. Yes our period increases to 4pi. Our function is y = sin (x/2).


Now, let's talk about shifts of our graph or when the sinusoidal function is moved horizontally or vertically. We see this when our wave is shifted up, down, to the left, or to the right. We call shifts to the left or right 'horizontal shifts' and shifts up and down 'vertical shifts.' If our wave is shifted 3 spaces to the right, we note this in our function by subtracting our shift from our variable. If our wave is shifted 3 spaces to the left, then we add this shift to our variable. So, a shift of 3 spaces to the right changes our function to y = sin (x - 3).

y = sin (x - 3)
sinusoidal function

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