# 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.