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
This resource is from Underground Mathematics.
Warm up
This is a graph of the function $f(x)=\dfrac{1}{x}.$
The graph has two asymptotes.
How would you describe what an asymptote is?
Main problem
Here are some descriptions or statements about asymptotes.
- "An asymptote is a line which a curve gets closer and closer to but doesn't meet."
- "An asymptote is a line which a curve approaches as $x$ tends to infinity."
- "A curve can't cross an asymptote."
- "Asymptotes are parallel to the coordinate axes."
- "A graph can only have one asymptote parallel to each axis."
- "Asymptotes occur when a function isn't defined for certain input values."
- "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. You might like to use Desmos to find possible equations for the curves shown in the cards.
A
B
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F
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N
O
P
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.
Student Solutions
Pushtoon has sent us a very nice solution, explaining why the statements given are not always true for asymptotes. You can see his solution here.
A group of year 12 Mathematicians from St Albans School explored this problem in one of their lessons. You can download their solutions using the link below.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
This algebra-free resource is designed to provoke discussion and help build up students’ sense of what asymptotes are.
Students are presented with a list of statements about asymptotes and a collection of sketch graphs. They can use the graphs and statements to help decide whether the graphs have asymptotes and whether the statements are good descriptions of what an asymptote is.
Note that in the descriptions of the features of the graphs, tangents are mentioned, but points of inflection are not mentioned.
Possible approach
Show students the warm-up and ask them what an asymptote is. Students’ descriptions of asymptotes, together with those suggested in the resource should be available for students to refer to when looking at the graphs.
Give students a few minutes to look through the graphs before starting to identify those with asymptotes. Students may find it helpful to draw in asymptotes, or at least see if there could be an asymptote by lining up a ruler. The resource is designed to promote discussion so students might start in pairs and then make small groups with another pair before a class plenary where they should be prepared to justify their position.
Once students have started to identify which graphs have asymptotes, you could ask them to start to note this on a board. If students say that a graph has asymptotes, it may also be interesting to ask them to say how many.
Students may find it difficult to give reasons why graphs do or do not have asymptotes. To support this, they could be asked to read through the explanations in the solutions for three graphs they find most interesting before having a plenary discussion on what they think an asymptote is.
Key questions
Can you justify your decision?
What might be true of the function where there are vertical asymptotes?
Do asymptotes have to be straight lines?