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Transforming the Tangent Graph - Problem 1

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.

I want to graph the transformation of the tangent function. I have this problem and it asks me to graph y equals 3 times tangent 1/4x. I’ve got the graph of tangent here and some key points on the table here. These are really important points to remember because if you know these, you can easily transform them and get a table of values for your new function and that’s just what I’m going to do now.

I want to show you a trick, whenever you’re transforming a function name the inside part theta, if you name the inside part theta, then theta equals 1/4x and that means x equals 4 theta. When you’re making your table of values for 3 and tangent ¼x the x values are going to be exactly 4 times theta. You don’t even have to think about transformations. This it makes it very easy. So I just multiply these guys by 4. I get -2pi, -pi, zero, pi and 2pi. That’s especially useful because horizontal transformations are the hardest to think about.

These 3 just means multiply the y values by 3. Now 3 times undefined is still undefined and those are going to give me my asymptotes. And then here, -1 times 3, -3, zero times 3, 1 times 3 and now I’ve got a table of data for one period of my function. Let’s go to the graph.

I’m going to make this 2pi and this -2pi. That’s where my asymptotes go. Let me draw those right away. So I’ve got my special asymptote pen, ruler. Once you have the asymptotes drawn for your first period you can actually see how long a period is, it’s 4pi and that means I’m going to have another asymptote every 4 pi. This is 4pi, 6 pi, I need another asymptote here, this is -4pi, -6pi and I need one here, I’ll draw 3 periods. Here’s another asymptote and here’s another one.

Now, I’m going to plot some points. I have these 3 points to plot and I’m going to plot these right in the center, -pi, -3, I’ll make this -3 right here and this is positive 3. So negative pi and negative 3 is right here, zero, zero and pi 3 and the great thing about the key points for a tangent graph is that they all line up so they’re really easy to duplicate. I’ll put them in here and I’ll put them and here and then we just draw a nice smooth curve. You have to know what the tangent graph looks like but it you know passes through the x axis at about a 45 degree angle and then curves upward, asymptotically approaching these line, so you got a curve like this, curve down, down, up and one more.

It’s really easy to draw a tangent function if you know these key points the two asymptotes and the points here. That’s the first step, write down the key points, second step, transform the points, third step graph a period and fourth step, fill in the additional periods, as many as you need. It’s really easy to graph transformations of tangents.

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