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High School Precalculus Textbook32 chapters | 253 lessons | 1 flashcard set

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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.

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:

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:

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:

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.

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.

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)*.

What if our wave is shifted up or down? If our shift is upwards, we add this shift to the end of our function. If our shift is downwards, we subtract this shift from the end of our function. For example, an upwards shift of 3 spaces is written as *y = sin (x) + 3*. Our vertical shift is written outside the argument of the sine function.

In math, we label our horizontal shifts, our left-right shifts, with the letter *C*, and we label our vertical shifts, our up and down shifts, with the letter *D*.

Putting it all together, we get this formula for our sinusoidal function that covers all possible changes to our wave:

Let's take a look at how we can use all this information. Remember, our *A* stands for our amplitude, the *B* stands for the change in our period defined by 2pi/B, and *C* stands for the horizontal shift, and the *D* stands for the vertical shift. Note here that our *B* has been factored out of the *x - C* part. Your *B* part may not always be factored out nicely like this in the problems that you see. If this is the case, you will need to factor out your *B* to find your *B* and *C*.

*Write a sinusoidal function that represents a wave that has been shifted 2 spaces to the right with a period of 4pi.*

From this problem, we see that the amplitude hasn't changed, so our *A* remains 1. The problem mentions that the wave has a period of 4pi. This means that our *B* must equal 1/2 for 2pi/(1/2) = 4pi. There is a horizontal shift of 2 spaces to the right, so our *C* is 2. There is no vertical shift, so our *D* is 0. We end up with this sinusoidal function:

All we did was plug in our values for our letters.

Let's review what we've learned:

A **sinusoidal function** is a function using the sine function. The basic form of a sinusoidal function is *y = A sin (B(x - C)) + D*, where *A* is the amplitude or height of our function, *B* is the change in period defined by 2pi/B, *C* the horizontal shift, and *D* the vertical shift. By plugging in our values for these letters, we are able to write a sinusoidal function that covers any change in our wave.

You will have the ability to do the following after watching this video lesson:

- Define sinusoidal function
- Identify the basic form of a sinusoidal function and each of its variables
- Explain how to write a sinusoidal function covering any changes in the wave

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High School Precalculus Textbook32 chapters | 253 lessons | 1 flashcard set

- Solve Trigonometric Equations with Identities & Inverses 5:44
- How to Solve Trigonometric Equations for X 4:57
- Properties of Inverse Trigonometric Functions 7:56
- Solving Trigonometric Equations with Restricted Domains 7:18
- Solving Trigonometric Equations with Infinite Solutions 6:32
- Finding the Sinusoidal Function 7:00
- Go to Solving Trigonometric Equations

- Go to Continuity

- Go to Limits

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