Asymptote = hyperbola?

  • Thread starter vanmaiden
  • Start date
  • #1
102
1

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?
 

Answers and Replies

  • #2
2,571
1
Do you even know what a hyperbola is...?
 
  • #3
102
1
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.
 
  • #4
2,571
1
Do you have a book?
 
  • #5
35,125
6,871
The graph of y = ln(x) has a vertical asymptote, but does not represent a hyperbola.
 
  • #6
102
1
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.
 
Last edited:
  • #7
102
1
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 :smile:
 
  • #8
SteamKing
Staff Emeritus
Science Advisor
Homework Helper
12,798
1,670
Well you have access to much more than a single textbook. You can use your computer and google 'hyperbola' or 'conic section'.
 
  • #9
Mentallic
Homework Helper
3,798
94
What about reciprocal functions of the form [tex]\frac{1}{x}[/tex] that have asymptotes at y=0,
or rational functions with a constant non-zero asymptote such as [tex]\frac{2x}{x+1}[/tex] or even an asymptote that is a not a line, [tex]\frac{x^3-1}{x}\approx\frac{(x-1)(x^2+x+1)}{x-1}=x^2+x+1, x\neq 1[/tex]. For this function as x gets very large positive or negative, the graph approaches the parabola [itex]y=x^2+x+1[/itex]
 
  • #10
35,125
6,871
What about reciprocal functions of the form [tex]\frac{1}{x}[/tex] 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 [tex]\frac{2x}{x+1}[/tex]
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, [tex]\frac{x^3-1}{x}\approx\frac{(x-1)(x^2+x+1)}{x-1}=x^2+x+1, x\neq 1[/tex]. For this function as x gets very large positive or negative, the graph approaches the parabola [itex]y=x^2+x+1[/itex]
 
  • #11
Mentallic
Homework Helper
3,798
94
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:
 
  • #13
102
1
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.
 

Related Threads on Asymptote = hyperbola?

  • Last Post
Replies
1
Views
12K
Replies
9
Views
1K
Replies
4
Views
3K
Replies
5
Views
9K
Replies
29
Views
1K
  • Last Post
Replies
7
Views
708
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
4
Views
2K
Top