PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
For graphing transformation as the cotangent function let's try a harder example. Graph y equals 2 cotangent one quarter theta minus 2. Now let’s start by listing some key points for our cotangent function.
I like to graph between 0 and pi, we have pi over 2 pi over 4 and 3 pi over 4. At 0 and pi cotangents undefined pi over 2 it's 0 at pi over 4 it's 1 and at 3 pi over 4 it's -1. So how’s that going to affect the graph of 2 cotangent one quarter theta minus 2?
Well I’ll make the substitution. U equals one quarter theta and that way theta equals 4u, so in order to find my values of theta I have to multiply these values by 4. So 0 times 4, 0, pi over 4 times 4, pi, pi over 2 times 4, 2pi 3pi and 4pi. Now to get my y values I have to take cotangent of u in the values I have here, multiply by 2 and subtract 2.
So 2 times undefined minus 2 it's still going to be undefined same thing here. But 2 times 1 minus 2 is 2 minus 2,0. 0 times 2 minus 2 -2. -1 times 2 -2, minus 2 is -4. So this will give me points that I can use to plot one good period of the cotangent graph.
Let me start by graphing the two vertical asymptotes x equals 0 and x equals 4pi. So x equals 0 is here right along the y axis, and x equals 4 pi is here. And it don’t matter I might as well graph other asymptotes. I know that these first two asymptotes define one period of my cotangent graph so the period is going to be 4 pi. That means I'll have asymptotes every 4pi so one at 8pi and one at -4pi as well. Now let me continue by graphing some key points between 0 and 4pi, these points. Pi, 0, 2pi,-2 and 3pi,-4.
Pi, 0, 2pi,-2 and 3pi,-4 and remember cotangent has a decreasing shape so kind of like this. So if I want to graph another period I just shift these three points over so this point goes over 4pi to here, this one goes to here, this one goes to here. Then I can shift them over this way as well I get one here, one here and one here then here we go.
We’ve got three periods of my cotangent graph notice it shifted down a little bit, the inflection points of the graph are below the x axis they are usually on it, but this is three periods of my function y equals 2 cotangent one quarter theta minus 2.