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.
Age
16 to 18
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

Powerful Quadratics


This resource is from Underground Mathematics.

 

 

Warm up



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

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

                                                                                

 

Approaching asymptotes
              
Approaching asymptotes
  

C                                                                             D

                                                                                 

Approaching asymptotes
                
Approaching asymptotes
 

E                                                                              F

                                                                                   

Approaching asymptotes
                         
Approaching asymptotes
  

G                                                                             H

                                                                                  

Approaching asymptotes
              
Approaching asymptotes
 

I                                                                              J

Approaching asymptotes
               
Approaching asymptotes
 

K                                                                            L

Approaching asymptotes
            
Approaching asymptotes
  

 



M                                                                            N

Approaching asymptotes
              
Approaching asymptotes
  



O                                                                            P

Approaching asymptotes
             
Approaching asymptotes



 

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