How do you find the points of an aysmptote graph?

Question

How do you find the points of an aysmptote graph?
I know how to find the slant asymptote but how do I graphy the curved functions part. For example. I know how to find the asymptote of 2x^2 + 1/ x but how do I get the points to graph the curves?

Answer 1

The easiest way is just to plug in numbers. For this one, you're going to have an asymptote at x = 0, and your graph is going to get really close to the vertical line at x = 0 on both sides. So plug in some numbers just to see what happens (I'm assuming that the 1/x term is by itself and the expression is NOT (2x^2 + 1)/x):

--Any negative numbers would give you a curve that goes negative very quickly. For x = -1, for example, y = 1, but when x = -0.5, y = -1.5, and for x = -0.25, y = -3.875.
--Any positive numbers would give you a curve that goes positive very quickly. For x = 1, y = 3, and for x = 0.5, y = 2.5, and for x = 0.25, y = 4.125

So from these you can see what happens right around the asymptote. As for the rest of the curve (away from the asymptote), again, the easiest way to find out what happens is to plug in some numbers (i.e. x = -2, x = -3, x = -4 for the left hand side, and similarly for the right hand side of the asymptote) and go from there.