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Transforming the Graphs of Sine and Cosine - Problem 3

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

I’m going to graph another transformation of cosine,one of these type y is equals a times cosine bx. As always I like to use my key points, so I’ve got those drawn out on the table and I’m going to make a second table for the function I’m about to graph. Here’s the problem.

Graph y equals 10 cosine pi x over 2. This one looks a little bit harder because there’s pi x over 2 but we’ll see that it’s really not much different from our previous examples we’ve done.

Let me write this out x, 10 cosine pi x over 12. Actually let me right that a different way. Let me right that as pi over 12 times x. The reason is it’s a little easier for me to see what kind of transformation this pi over 12 is going to give me.

Now that’s the next step, figuring out what kind of transformations I have here, so I can figure out how to transform this data. This is pretty easy, the 10, the multiplication by 10 out in front means I’ve got a vertical stretch by a factor of 10. So all the y values of a regular cosine graph are going to get multiplied by 10, so I can write that down immediately. 1 times 10, 10, 0 times 10, 0, -1 times 10 -10, 0, and 10.

But what does this represent? Usually the coefficient in front of the x gives me a horizontal compression or stretching. But pi over 12 is kind of a strange number now just to give you an idea pi is about 3 so this is about 1 quarter that means I've got a horizontal stretch but that’s not super helpful. If I had a stretch, a horizontal stretch of 4 rather, I would multiply these values by 4 and I would have a one quarter here. So the key is I’ve got to multiply the x values by the reciprocal of this in order to get the new x values. So I’m going to multiply each of these values by 12 over pi.

So 0 times 12 over pi is 0, pi over 2 times 12 over pi, pi’s cancel 6 pi times 12 over pi is 12. 3 pi over 2 times 12 over pi you can probably guess it's 18 and 2 pi times 12 over pi 24. And so now you can kind of see since this is in period of cosine. One period of this graph is going to be 24, period will be 24.

Let’s plot this data and we’ll get one nice period of y equals 10 cosine pi over 12x. I’m going to make this value 10, -10 and I’m going to make this value up here 24. That makes this 12, this -12 and this -24.

Let me start with (0,10). It won’t matter I might as well graph 24, 10 I’m sorry yeah 24, 10. (6,0) this point (12,-10), (18,0). So these five points will give me one nice period of my new graph. And so if I want to get more than one period, I just duplicate these points over here, so I kind of shift everything over.

So I’ll come back down to here then down to -10 again back up to 0 and back up to 10. And I just connect those points and I’ll have a second period. And that’s y equals 10 cosine pi x over 12. Now what about the amplitude and period? Remember the formulas; amplitude is the absolute value of a. Now a in this case is 10 so the absolute value of a is 10. Period is 2 pi over b, b in this case is pi over 12, so 2 pi over pi over 12 is 2 pi times 12 over pi. When you are dividing fractions, you multiply by the reciprocal the pi’s cancel and you are left with 24.

As we noticed before, the period is 24. So when you are graphing the transformation of cosines remember, use your key points and transform them, graph the transformed points and duplicate as much as you need to to get the number of periods your teacher wants. Then for amplitude and period remember the formulas, amplitude is the absolute value of a period is 2 pi over b.

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