Asymptote = hyperbola?

1. Aug 26, 2011

vanmaiden

1. The problem statement, all variables and given/known data
If a graph has an asymptote, does that mean it's always going to be a hyperbola?

2. Relevant equations

3. The attempt at a solution
Well, I started to think of y=tan(x) and y=cot(x). I believe they would be called trigonometric circular functions as they repeat, but are they still considered hyperbolas because they have asymptotes?

2. Aug 26, 2011

flyingpig

Do you even know what a hyperbola is...?

3. Aug 26, 2011

vanmaiden

To be honest, my understanding of them is not that strong. You caught me. This is why I am asking.

4. Aug 26, 2011

flyingpig

Do you have a book?

5. Aug 26, 2011

Staff: Mentor

The graph of y = ln(x) has a vertical asymptote, but does not represent a hyperbola.

6. Aug 26, 2011

vanmaiden

Yep, I have calculus books. Unfortunately, they don't go into hyperbola's hardly at all. The one I have for school just deals with parabolas.

Last edited: Aug 26, 2011
7. Aug 26, 2011

vanmaiden

Hey, thank you. That's just what I needed

8. Aug 26, 2011

SteamKing

Staff Emeritus

9. Aug 26, 2011

Mentallic

What about reciprocal functions of the form $$\frac{1}{x}$$ that have asymptotes at y=0,
or rational functions with a constant non-zero asymptote such as $$\frac{2x}{x+1}$$ or even an asymptote that is a not a line, $$\frac{x^3-1}{x}\approx\frac{(x-1)(x^2+x+1)}{x-1}=x^2+x+1, x\neq 1$$. For this function as x gets very large positive or negative, the graph approaches the parabola $y=x^2+x+1$

10. Aug 26, 2011

Staff: Mentor

This is a hyperbola. The central axis is rotated by 45°.

This is the same as 2 + (-2)/(x + 1), so this is just the translation and stretching of y = 1/x, so is also a hyperbola.

11. Aug 27, 2011

Mentallic

Oh yes of course, why did my mind instantly jump to the general form of a hyperbola...?

Yes I'm aware of the second example's translations, but I don't really see why I bothered mentioning it now that you brought it up.

Clearly my brain's still in holiday mode :zzz:

12. Aug 27, 2011

Staff: Mentor

Happens to us all...

13. Aug 27, 2011

vanmaiden

Same here, man. Same here. Just started senior year.

True, I have been doing research on the internet. Every time I get on a site, it just wants to talk about those hyperbola's symmetric along the x or y axis. I have a decent understanding of them, just not these like 1/x lol.