Rational functions: asymptotes and graphs

Summary

A rational function is the quotient of two polynomials. This lesson teaches how to study it and how to sketch its graph with two algebraic tools: the sign chart and polynomial division. The zeros of the denominator determine the domain and give, according to whether or not the numerator vanishes at them, removable points or vertical asymptotes, near which the sign chart decides where the curve escapes. Division writes the function as a polynomial plus a fraction that tends to zero, and that polynomial is the horizontal, oblique or curvilinear asymptote that the graph approaches when the variable tends to infinity. The intercepts complete the sketch, and the range is obtained by solving for the variable when this is possible or, otherwise, from the behavior of the branches.

Learning objectives

Determine the domain of a rational function and distinguish, among the zeros of the denominator, the removable points from the vertical asymptotes.
Describe the behavior of a rational function on each side of a vertical asymptote by means of its sign chart.
Obtain, by means of polynomial division, the horizontal, oblique or curvilinear asymptote of a rational function and the position of the graph relative to it.
Sketch the graph of a rational function and determine its range, exactly when the equation $y = f(x)$ can be solved and, otherwise, to the extent that its branches allow.
Model applied situations with rational functions, such as the power delivered by a battery, the average cost of a production run or the concentration of a mixture, and interpret their asymptotes and their range in context.

Rational functions and their domain

In the lesson on algebraic functions, a function was called rational if it is the quotient of two polynomials, with a denominator that is not the zero polynomial:

f(x) = \frac{P(x)}{Q(x)}

Its domain consists of the real numbers that do not make the denominator vanish. Finding it therefore requires finding the roots of $Q$ , and for this one can use the quadratic formula and the techniques of the lesson on factoring quadratic polynomials, together with the remainder and factor theorems of the lesson on the algebra of polynomials. Every study of a rational function begins, consequently, by factoring its denominator.

Now, not all the zeros of the denominator play the same role. As was seen in the lesson on sign charts and inequalities, if a number $r$ makes both $Q$ and $P$ vanish, then by the factor theorem both polynomials are divisible by $x - r$ , and the common factor cancels at every point of the domain. If the denominator of the simplified fraction no longer vanishes at $r$ , the number $r$ is a removable point: the graph of $f$ is that of the simplified fraction with a single point removed, and the function would be defined at $r$ if it were assigned the value of the simplified fraction. If, on the contrary, after all the common factors have been cancelled the number $r$ still makes the denominator vanish, the line $x = r$ is a vertical asymptote, as is made precise below. That is why it is advisable to simplify the fraction before studying it, without forgetting that the cancelled numbers remain outside the domain.