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Intercepts and Asymptotes of Tangent Functions - 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.

Sometimes on your homework, you’ll be asked to find the x intercepts and asymptotes of a tangent function. Let’s find it for y equals -2 tangent of 5x. Remember the -2 is not going to affect asymptotes or x intercepts because it’s a vertical stretch and then a reflection, it’s this guy that affects the asymptotes and intercepts so let’s call this theta.

Remember it’s when theta is an integer multiple pi that tangent equals zero. So where n is an integer, now that means that 5x is an integer multiple of pi. Divide both sides by 5 and you get n pi over 5. These would be the points where this function equals zero, so the intercepts would be, for example when n is one, pi over 5 zero, when n is 2, 2pi over 5 zero, 3pi over 5 zero and so on. Those would be the intercepts.

What about the asymptotes? Remember tangent is undefined when theta equals pi over 2 plus n pi. Right now theta is 5x. Again we divide by 5 and we get the values of x where this function’s undefined. Pi over 10 plus n pi over 5 and by the way n pi over 5 is the same as 2n pi over 10. That will help us calculate some values. So for example the asymptotes would be, and these are the vertical asymptotes, x equals, when x is zero, pi over 10, when n is 1 you get pi over 10 plus 2 pi over 10, 3 pi over 10. When n is 2 this is 4pi over 10 plus pi over 10, 5pi over 10 and so on. And it goes in both directions.

The asymptotes would be pi x equals pi over 10, x equals 3pi over 10, x equals 5pi over 10 and so on. These are all multiples of pi over 10 and here the intercepts, pi over 5 zero, 2pi over 5 zero, 3 pi over 5 zero, integer multiples of pi over 5.

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