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Transforming the Graphs of Sine and Cosine - Problem 1

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.

When graphing transformations of sine and cosine right now, we are focusing on transformations of this form. Y equals a sine b x. So here is a problem. Graph y equals -3 sine 1/2x.

The first thing I want to do, oh it also asked me that identify the amplitude and period, I’ll do that a little but later the first thing I want to do is recall the key points of the sine graph.

Theta sine theta, the key points are the points where the values are 0, 1 or -1. So you know that sine goes with point (0,0) pi over 2, 1 pi 0, 3 pi over 2, -1 and 2 pi 0. These are my five key points this will give me one period of the sine graph and I can get more period just by sort of copying.

My graph of x -3 sine 1/2 x this is a transformation of the sine graph and I need to figure out what kind of transformation. The -3 is like a vertical stretch, it’s also a reflection across the x axis and what the -3 does is basically multiplies the y values of sine by -3. So I can take these y values and multiply them by -3 and get new y values. This part the 1/2x, this is a horizontal stretch. Remember when this value is bigger than 1 we actually get a compression, when this value is between 0 and where we get a stretch. So this is going to give me a stretch by a factor of 2. That’s really useful because I can take these values and just multiply them by 2.

So let me do that. These values times 2, 0 pi, 2 pi, 3 pi, 4 pi and then these values multiplied by -3. 0, -3, 0,3 and 0. And this is my set of coordinates for my new graph, all I have to do is plot these. So let me start with (0,0). Now I need some kind of labelling system here so I know how big the quantities are. This is going to be 3, I’ll make this pi, 2 pi, 3 pi, 4 pi. Pi -3 is down here, 2 pi 0 and 4 pi 0, let me just plot those right now. That means that 3 pi we’ve got 3. So that gives me this graph.

And if I want to graph 2 periods or 3 periods of any number periods, I just copy these points because this is a four period of my new graph, so I’ll work backwards. I have a point here and here then here let’s say here and I can continue forward too if I want to. So that’s two periods of y equals -3 sine 1/2x.

Now the amplitude and period remember the formulas we’ve discussed before, amplitude is the absolute value of a where a is this coefficient right here. The absolute value if -3 is 3, and the period is 2 pi over b. Period 2 pi over b, b is this coefficient so we have 2 pi over 1/2 and that’s 4 pi because you can see that in your graph you can see that you’ve got a period of 4 pi.

So remember when you are graphing stretches and shrinks of the sine graph or the cosine graph start with the key points transform the points first then plot them. The period and amplitude are easy just remember the formulas.

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