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.