Approaching asymptotes

Can you describe what an asymptote is? This resource includes a list of statements about asymptotes and a collection of graphs, some of which have asymptotes. Use the graphs to help you decide whether you agree with the statements about asymptotes.

Problem

Image
Powerful Quadratics



This resource is from Underground Mathematics.

 

 

Warm up



This is a graph of the function $f(x)=\dfrac{1}{x}.$

 

Image
Approaching asymptotes

 

The graph has two asymptotes.

How would you describe what an asymptote is?

 

Main problem



Here are some descriptions or statements about asymptotes. 

  1. "An asymptote is a line which a curve gets closer and closer to but doesn't meet."
  2. "An asymptote is a line which a curve approaches as $x$ tends to infinity."
  3. "A curve can't cross an asymptote."
  4. "Asymptotes are parallel to the coordinate axes."
  5. "A graph can only have one asymptote parallel to each axis."
  6. "Asymptotes occur when a function isn't defined for certain input values."
  7. "A function tends to positive infinity on one side of an asymptote and tends to negative infinity on the other side."



Here are some examples of curves, some of which have asymptotes.  Use these to help you decide whether you agree with the statements above. Printable versions of these cards can be downloaded here

A                                                                             B

                                                                                

 

Image
Approaching asymptotes

              

Image
Approaching asymptotes

  

C                                                                             D

                                                                                 

 

Image
Approaching asymptotes

                

Image
Approaching asymptotes

 

E                                                                              F

                                                                                   

 

Image
Approaching asymptotes

                         

Image
Approaching asymptotes

  

G                                                                             H

                                                                                  

 

Image
Approaching asymptotes

              

Image
Approaching asymptotes

 

I                                                                              J

 

Image
Approaching asymptotes

               

Image
Approaching asymptotes

 

K                                                                            L

 

Image
Approaching asymptotes

            

Image
Approaching asymptotes

  

 



M                                                                            N

 

Image
Approaching asymptotes

              

Image
Approaching asymptotes

  



O                                                                            P

 

Image
Approaching asymptotes

             

Image
Approaching asymptotes



 

This is an Underground Mathematics resource.

Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.

Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.