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Transforming the Tangent Graph - Concept

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 a tangent transformation, start by using a theta and tan(theta) t-table for -pi/2 to pi/2. In the case of y=Atan(Bx) or y=Atan(B(x-h)), define Bx or B(x-h) to be equal to theta and solve for x. Now use this equation to create a x and Atan(Bx) or Atan(B(x-h)) table, which will give coordinate pairs to plot.

The previous episode, I graphed the tangent function between 0 and pi over 2. Now I want to extend that graph to the left and to the right, and I'm going to need some facts from before. First of all I have a table of values that I found before. Tangent of 0 is 0, tangent of pi over 4 is 1 and tangent of pi over 2 is undefined. And this is what gives us that vertical asymptote at x equals pi over 2. I also need 2 facts. The identity tangent of -x equals negative tangent x and that tangent has period pi. Now this this fact means that if you take at 2 inputs that have opposite values you get opposite outputs.
So, for example if I plug negative pi over 4 into tangent, I get the opposite output -1. For tangent's an odd function. Of course the opposite of 0 is 0 gives me the opposite output which is also 0 and the opposite input negative pi over 2, it's just it's still going to be undefined. And this helps me complete a single period of tangent functions. Let's go back here and pi over 4, I'm going to have -1 and I'm going to have an asymptote at negative pi over 2, just like I have with pi over 2. So let me draw that really quickly. And I complete the graph like so. Now this gives me one period of tangent because the period of tangent is pi. So I want to duplicate this here and here, at least 2 more periods.
Now let's observe that the key points that I have here, negative pi over 4 -1, 0 0, and pi over 4 1, all line up. So it's really easy to draw them over and over again. They would go here and here. And once you've got them drawn it's super easy to draw the tangent curve. Just remember the shape. And then you can put your asymptotes on. There are going to be asymptotes every pi units. This asymptote is at 3 pi over 2, this one's at -3 pi over 2. That's the great thing about periodic functions because they're periodic, once you have one cycle drawn you just duplicate that cycle as many times as you need to.
So here I have 3 full periods of the tangent function and now that I have that, I can actually apply transformations to the tangent function which I'll do in future episodes.

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