The purpose of this lesson is to introduce students to vertical asymptotes. The line \(x=a\) is a vertical asymptote for a rational function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. Students begin by reasoning about a vertical asymptote of a simple rational function that represents the relationship between the time and speed needed to travel a fixed distance, building on the work they did in the previous lesson around a cylinder of fixed volume. From there, students complete a card sort in which they match equations and graphs of rational functions, focusing on making connections between the structure of the two representations (MP7) and analyzing representations and structures closely (MP2).
While the end behavior of rational functions is touched on here as part of making sense of a context, the following lesson investigates end behavior and horizontal asymptotes in more depth. As such, only a light touch is needed on these ideas in this lesson, with an emphasis on adapting the previously established language around end behavior of polynomials to fit specific rational contexts.
- Identify features of simple rational functions from graphs and equations.
- Interpret the end behavior of a rational function in context.
- Let’s explore graphs and equations of rational functions.
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)
- I can identify a vertical asymptote from a graph or an equation of a rational function.
A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.
The line \(x=a\) is a vertical asymptote for a function \(f\) if \(f\) is undefined at \(x=a\) and its outputs get larger and larger in the negative or positive direction when \(x\) gets closer and closer to \(a\) on each side of the line. This means the graph goes off in the vertical direction on either side of the line.