# Asymptote = hyperbola?

## Homework Statement

If a graph has an asymptote, does that mean it's always going to be a hyperbola?

## 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?

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

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

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

Do you have a book?

Mark44
Mentor
The graph of y = ln(x) has a vertical asymptote, but does not represent a hyperbola.

Do you have a book?

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.

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

Hey, thank you. That's just what I needed SteamKing
Staff Emeritus
Homework Helper

Mentallic
Homework Helper
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 $y=x^2+x+1$

Mark44
Mentor
What about reciprocal functions of the form $$\frac{1}{x}$$ that have asymptotes at y=0,
This is a hyperbola. The central axis is rotated by 45°.

or rational functions with a constant non-zero asymptote such as $$\frac{2x}{x+1}$$
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.
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 $y=x^2+x+1$

Mentallic
Homework Helper
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:

Mark44
Mentor
Happens to us all...

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

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