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Transforming the Tangent Graph - Problem 2

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.

We’re graphing transformations of tangents. I want to graph y equals negative ½ tangent of pi x. Whenever I’m graphing transformations, I like to sometimes make a little substitution theta equals pi x because it helps me understand how the horizontal transformations work.

Let me go back here, I’ve got my key points for tangent written out on a table. They are three points and these two undefined points actually are very helpful because they tell me where there vertical asymptotes are. I’ll write up here, theta equals pi x. Now on these table I want to make points negative ½ tangent pi x. I want to plot out, I want to list the points for this graph and so I have to transform these points to here using the transformations.

Now if theta equals pi times x then x equals 1 over pi times theta. In order to get my x values I have to take theta values and multiply by 1 over pi. This minus pi over 2 times 1 over pi is -½ minus ¼ times 1 over pi, -1/4, 0, ¼, ½ and then this -1/2 in front means multiply the y values here by -1/2. Undefined times -1/2 is still undefined, times -1/2 is positive ½, zero, -1/2, still undefined. My transformed graph has this table data, two asymptotes that I can plot right away in, -1/2 and ½. On my graph I’m going to make this -1/2 and this ½. Let me draw the asymptotes right away.

And now I’ve got two asymptotes, these two asymptotes bound one period of the tangent function. This also tells me what the period of the tangent function, it’s 1. That the distance between -1/2 and 1 so if I want to find out what the other asymptotes are I just need to fill in the rest of this. The other asymptotes will be one unit away so one unit away from ½ is 3/2 and one unit away from negative ½ is 3/2. That’s enough asymptotes to draw three periods of the tangent function.

I take my key points and I plot them. Negative ½, ¼, ½. Let’s make this ½ and this -1/2, -¼,½ will be here and I’ve got 0,0 and 1/4,-1/2. And the key points always line up which is really nice so they’re easy to duplicate let me do that now. Key points here and then over here and then over here and the just draw a nice tangent curve in the second period and then the third period, missed a point there, that’s okay. So we have three periods of our tangent graph.

Remember the key steps, write the key points, transform the key points on a table, draw a period and then duplicate the periods.

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