Asymptotes and Hyperbolas: Exploring the Relationship

[vanmaiden]

Homework Statement

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

Homework Equations

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?

 

[flyingpig]

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

 

[vanmaiden]

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.

 

[flyingpig]

Do you have a book?

 

[Mark44]

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

 

[vanmaiden]

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.

 

[vanmaiden]

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]

Well you have access to much more than a single textbook. You can use your computer and google 'hyperbola' or 'conic section'.

 

[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 $y=x^2+x+1$

 

[Mark44]

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]

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]

Happens to us all...

 

[vanmaiden]

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

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

Well you have access to much more than a single textbook. You can use your computer and google 'hyperbola' or 'conic section'.

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.