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More Transformations of Sine and Cosine - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
I want to graph another transformation of cosine. Here I'm asked to graph y equals -2 cosine of pi over 2x plus pi over 2. Now in order to completely understand the transformations here I'm going to have to factor the pi over 2 out, so let me do that.
-2 cosine pi over 2 times the quantity. If I pull pi over 2 out of this term, I'm left with x and if I pull it out of this term, I'm left with 1. So this means I have a shift to the left, one unit. Remember this is of the form x minus h, in this case h is -1 and h is our phase shift.
Now I usually apply the phase shift last, before I do that usually I graph the un-shifted version of this which is y equals -2 cosine of pi over 2x. I start out with my key point, my cosine key points here I'm going to transform these and get a table of my stretched or shrunk graph and then I'll graph that and shift it.
So what do these numbers tell me about the transformations of cosine? The -2 is a reflection across the x axis and a stretch by a factor of 2 and it affects the y values of the cosine. So these guys are multiplied by -2, 1 becomes -2, 0 becomes 0, 2, 0 -2. The pi over 2 is going to be a horizontal compression or stretch and the factor of the stretch or compression is the reciprocal of pi over 2, it's 2 over pi. I don't even need to know what number that is, I just need to know that I have to multiply these values by 2 over pi.
So 0 times 2 over pi is 0, pi over 2 times 2 over pi is 1, pi times 2 over pi is 2, 3 pi over 2 times 2 over is 3 and 2 pi times 2 over pi, 4. And so you can see that the whole period of 2 pi has been transformed into a new period of 4, I'll come back to that in a minute. Let me graph these five points, 0,-2 so I'm making this -2 and this 2 let's make this 4. We'll graph the zeros, 1,0 and 3,0, 2,2 and finally 4,-2 let me connect these.
I'm going to draw the second period to the left so connecting it backwards, so these points are just these points shifted over one whole period let me fill this out and that gives me my un-shifted cosine curve y equals -2 cosine pi over 2x. So now I have to draw this shifted curve.
Pulling back here let me remind myself the phase shift was -1, so my new graph is going to be the graph I just drew shifted to the left one unit, let's do that. So all of these points need to be shifted to the left one unit. This point is going to go here, this point's going here, this point's going here, here, here, here, here, here, and here and then I just draw a smooth curve connecting these and the result is my final graph this is y equals -2 cosine of pi over 2 times x plus 1 the quantity.
That takes us to the last thing amplitude and period. The amplitude of this is the absolute value of A, well A is the coefficient in front of the cosine. The absolute value of -2 is 2 and the period we already found that the period was 4, but remember the formula that will verify what our period is. 2 pi over b and b is this coefficient here. 2 pi over pi over 2 which is the same as 2 pi times 2 over pi. Pi's cancel and you get 4. The period is 4, amplitude is 2, phase shift is -1 and these are all two graphs.
Remember when you draw more than one graph, you should probably label them so that your teacher knows which one is which. Just to review, key points stretch the points, make a table of values, plot the stretched graph and then finally do your shift.
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