PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Graphing transformations of the cotangent function, here is another one y equals cotangent of pi over 4 theta plus pi over 4. Now let’s recall the key points for the tangent function. I’d like to graph between 0 and pi how I did pi over 2, pi over 4, and 3 pi over 4.
Now recall that at 0 and pi cotangent is undefined. pi over 2 it's 0, the pi over 4 it's 1, and 3 pi over 4 it's -1. So these are the key points I’d like to start with. Now from my transformed function I'm going to need to make a substitution, I’m going to substitute u for pi over 4 theta plus pi over 4. Now that will tell me what I need to do to u in order to get theta. First let’s subtract pi over 4 from both sides. And then I'll multiply both sides by 4 over pi. I’ll get 4 over pi u minus 1 equals theta.
So I need to take my u values multiply by 4 over pi and then subtract 1 and that will give me my theta values. So let’s start with 0, 0 times 4 over pi is 0, minus 1, -1. Pi over 4 times 4 over pi is 1 minus 1 is 0. Pi over 2 times 4 over pi is 2, minus 1 is 1, 3 pi over 4 times 4 over pi is 3 minus 1 is 2 and you kind of see the pattern this is going to be 3.
Now what do I do with cotangent of u? It actually doesn’t look like I do anything, this is exactly cotangent use, I just copy this value down. Undefined 1, 0 -1 undefined.
Now the undefined points tell me where the vertical asymptotes are; -1 and 3 and they also tell me how long a period is, a period is going to be 4 units from -1 to 3. So let me plot my vertical asymptotes, -1 and 3. And I might as well plot a couple others 4 units to the left, -5 actually that’s all it fits so let me just stop there. And then I’ll plot my points I’ve got 0,1, 1,0 and 2,-1. And if you recall the shape of the cotangent graph it’s a decreasing version of the tangent graph so it looks something like this.
And then if we want to graph more periods we just translate these points to the right or to the left by 1 full, so 4 units to the right this point becomes 5,0 points to the right this point becomes 4,1, this one becomes 6,-1. So I get something like this then I can shift this to the left as well.
There you have three periods of the function y equals cotangent pi over 4 theta plus pi over 4.