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The prompt
The prompt
Mathematical inquiry processes: Explore and generate examples; test different types of cases; generalise; analyse structure and reason. Conceptual field of inquiry: graphs of rational functions; asymptotes.
Even though the prompt is true, it can lead students to make a false generalisation. Once they have verified the statement, students have been quick to declare that the graph of a rational function has the same number of asymptotes as it has binomial expressions.
Moreover, they might add that the number of expressions in the denominator equals the number of vertical asymptotes, while the number of expressions in the numerator equals the number of horizontal asymptotes. The rules 'work' when graphing the functions in the prompt (see graphs below).
However, when students begin to change the prompt, they realise that matters are not so simple. The graph of the equation with two expressions in the numerator (below), for example, has one vertical asymptote and one slant (or oblique) asymptote.
Question, notice, and wonder
Question, notice, and wonder
In the question, notice, and wonder phase of the inquiry, students have made the following contributions:
What is an asymptote?
An asymptote is a straight line that a curve approaches but never intersects.
It is a line where the graph of an equation is undefined.
When you draw the graph of the first equation on Desmos, there are two asymptotes.
The constants in the expressions increase by one each time.
The coefficient of x is one in each expression. What would happen if the coefficient is greater than one?
Are the number of asymptotes linked to the number of expressions in the functions?
Discovery or inquiry?
Discovery or inquiry?
Inquiry Maths is not a discovery model of learning. When students need a new concept or procedure to make progress in a line of inquiry, they can look it up or draw on the knowledge of the teacher. Students might discover an interesting or even novel result, but that is not the aim of the inquiry. Rather, mathematical inquiry involves exploration, conjecture, generalisation, and proof.
In the rational functions inquiry, students might discover the rule for vertical asymptotes by exploring different graphs. However, the rules for horizontal asymptotes are more difficult to identify by examining different cases.
The teacher might consider the rule for vertical asymptotes as 'proof of existence' and either guide students to research the rules for horizontal asymptotes or provide them directly. The inquiry then becomes about using the rules to sketch graphs from functions and explaining, or even proving, them.
Asymptotes and holes
Asymptotes and holes
The distinction between a vertical asymptote and a hole in the graph is often not immediately obvious during independent exploration. Students might divide the numerator and denominator by a common factor without realising that the factor affects the graph. For example, if the factor is (x - 4), the graph will not be defined at x = 4. When students hover the cursor over the graph on Desmos, they see the hole.
November 2026
Lines of inquiry
Lines of inquiry
The slides contain lines of inquiry based on six regulatory cards.
In a guided inquiry, students suggest a course of action by selecting a card and justifying their selection to the class.
For the meaning of each card in the context of the inquiry, see the descriptions below.
Students might start the inquiry by graphing the functions on Desmos. They can infer the meaning of 'asymptote' from the graphs, if they do not already know it, and verify the prompt is true. They might notice the connection between the expressions in the denominator and the equations of the asymptotes.
Students generate more examples of the same two types of rational functions to determine whether they always have, respectively, two and three asymptotes.
The inquiry moves onto different types of functions (see a selection below). The teacher might introduce the concept of the degree of the polynomial and suggest that students start to consider the horizontal asymptote in terms of the degree of the numerator and denominator.
As students explore they make conjectures about the connections between rational functions and their graphs. They test and amend the conjectures until they are confident to form generalisations.
Throughout the inquiry, the teacher encourages students to explain the reasons behind their conjectures.
Analysing the mathematical structure of a function provides the basis of an explanation. For example, when the denominator equals zero the function is undefined because division by zero is undefined in standard arithmetic. Thus, for the first case in the prompt, the function is not defined when x + 2 = 0. Therefore, x = -2 is the equation of the vertical asymptote.
See the slides for more information on horizontal asymptotes.
A proof that the horizontal asymptote is y = 0 when the degree of the numerator is less than the degree of the denominator (see beloe) can serve as a model as students attempt to prove the other cases for horizontal asymptotes. (See the proofs in the slides)
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