Question

How might being introduced to vertical asymptotes graphically, with no algebraic work, help or hurt one's understanding of the topic? How does looking at graphs of rational functions, created by graphing utilities, enhance one's understanding of the algebraic work on the rational equations? How does creating graphs by hand based on the equation properties enhance your understanding?

Answer 1

I don't think it will hurt much to explore these topics. But if you really want to get a good grasp on asymptotes then you owe it to yourself to dig into the topic of Limits (some of the algebraic work ;D).

Take horizontal asymptotes for example. Limits will tell you what the graph is doing at the far far left and right ends of that function. As x gets bigger and bigger and bigger, am I getting closer to a specific value (an asymptote), or does my function just continue to grow infinitely large as time progresses.

Vertical asymptotes are a little different since they don't happen way out at the ends, but rather right in the middle of the actions.

Maybe you have a function like f(x) = x^2 / (x-1)
And you want to know what happens as you move from 0 to 2 along the x-axis. Welllll you have a problem, the graph isn't defined at x=1.
So taking the limit, getting closer and closer to 1, will tell you what the function is doing at that point.

And you might say to yourself, well it's undefined there, so it's obviously just going to blow up and approach infinity.
But is it approaching positive or negative infinity? Is it exploding upward or downward? It will do different things depending on which DIRECTION you approach x=1 from.

Are you familiar with Trig at all?
Because looking at the Tangent function is another nice function to try and wrap your head around.

Ok just some food for thought! :)

Answer 2

Thank you SO much! I'm a total dunce with it comes to anything concerning math. :) I am very grateful for your help.