Graphing Sine and Cosine Transformations

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  • 2:17 Amplitude Transformations
  • 3:12 Period Transformations
  • 4:30 Phase Shift Transformations
  • 5:30 Sample Problem #1
  • 6:30 Sample Problem #2
  • 7:42 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

What happens when we take the sine and cosine wave and transform them? They get bigger, smaller, faster and slower. Find out how in this lesson on sine and cosine wave transformations.

Ride the Waves

Let's say we go surfing. Actually, no, let's say you go surfing. I'm not going to be shark bait. But you? You're less terrified of the ocean than I am.

So, you're sitting there on your board, waiting for the perfect wave to ride. You notice that the waves move in regular patterns. There's this up and down that is remarkably consistent. This wave motion is what we're going to explore here.

By the way, I see the same thing from the safety of the shore. And as much as I love math, I'm not risking death by shark for it. Good for you, though.

The Sine and Cosine Wave

When you're sitting there, those even, steady waves are like a sine or cosine wave. The sine wave is the graph of y = sin x. It looks like this:

The sine wave
sine wave

It starts at (0,0) and moves up and down based on the values of sine. The sine of pi/2 is 1, so our graph hits 1 there. The sine of pi is 0, so it's back to 0 there. At 3pi/2, it's -1, then back up to 0 by 2pi, which is one full circle. 2pi is just 360 degrees in radians. That single up and down wave is called a period. Like the waves in the ocean, it keeps going. But we call one full revolution a period.

Then there's the cosine wave. If you've ever surfed, and you know I haven't, you know that the waves are different on different days. Trigonometry is no different. Here's the graph of y = cos x, or the cosine wave:

The cosine wave
cosine wave

When x is 0, the cosine is 1, so instead of starting at (0,0) like the sine wave, the cosine wave starts at (1,0). Then it goes down until we hit pi, where the cosine is -1, before going back up to 1 at 2pi.

Again, the period is 2pi. There's also the amplitude. This is the largest distance the wave travels from 0. That's going to be 1 for both the sine and cosine waves. Note that the amplitude is just the distance from 0, not the total vertical distance. It's like the height of a wave from the horizon, as opposed to from the dip it makes.

Amplitude Transformations

So, that's y = sin x and y = cos x. But, what if our equation is a little more complicated? What if we're on a surfboard - I'm sorry, what if you're on a surfboard - and the sea is angry that day? Bigger waves, right?

We can transform our graph by using this equation: y = Asin(Bx + C). This is really just bells and whistles. y = sin x is the same as this equation, but in this equation, A and B are 1 and C is 0. But what if they're not?

Let's start by changing A. What happens with y = 2sin x? Basically, our y values go twice as far as with y = sin x. So, the A impacts the amplitude. A for amplitude! Neat. So instead of 1, as with y = sin x, the amplitude of y = 2sin x is 2.

Period Transformations

Okay, I haven't seen any sharks yet, just ordinary waves and... Wait! What's that? Oh, just some driftwood. Yeah, I'm lots of fun at the beach.

Anyway, let's next manipulate B. Let's try y = sin 2x. What happens to our graph? Well, we still start at (0,0). But now we go up and down in a shorter period. That graph looks like this:

The period of this graph is pi.
period transformation graph

Why? If we plug in values for x, we see that instead of hitting 1 at pi/2, we hit it at pi/4. So, our period isn't 2pi, it's pi.

That means that our B value affects the period of the graph and that relationship is 2pi/B. Whatever our B is, just plug that into 2pi/B, and you'll have your new period. So, our graph of y = sin 2x shows a much choppier ocean, with faster waves. This is when it's harder to see sharks. By the way, the sand on the beach, on good old terra firma, is nice and calm today.

How do we remember that B changes the period? Hmmm, B for period if you sort of mumble 'beriod?' We used the sine wave in these examples, but the facts are no different with the cosine wave.

Phase Shift Transformations

Finally, what about C? Remember, C was 0 in y = sin x. What about y = sin(x + pi)? This is going to bump our graph forwards or backwards along the x-axis. This is called a phase shift. The phase shift can be defined as -C/B. In this equation, -C/B is -pi/1, or just -pi.

Now, phase shift is a little trickier to remember, but you can always just plot some points to figure it out. In our example below, what happens when x is 0? We still have that pi there, so it's the sine of pi. Where is that on the graph? Where the middle arrow is. So, our wave is crossing the x-axis at 0.

Example of phase shift
phase shift

But see how it shifted? Our wave shifted back the distance of -C/B. So, we backed up our waves. You don't usually see waves going backwards, do you? If you do, you're probably surfing the wrong way.

Sample Problem #1

I bet you know what's coming next. Let's see what happens when we put it all together. What is the graph of y = (1/2)sin(4x - pi)?

Let's use what we know. Our A is 1/2. A is amplitude. So, our amplitude is 1/2. That means the graph will peak at 1/2.

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