Rational Functions

Rational functions are ratios of polynomials, and their graphs can show dramatic behavior such as sharp turns, breaks, and values that grow without bound. They are important because they model many real systems, including rates, inverse relationships, and situations with restrictions on allowed inputs. To read these graphs well, students need to identify asymptotes, holes, intercepts, and the overall shape of each branch. These features reveal where the function is undefined, how it behaves near critical x-values, and what happens far from the origin.

A rational function often becomes easier to analyze after factoring the numerator and denominator. Shared factors can create holes, while noncanceling denominator factors create vertical asymptotes. Comparing the degrees of the numerator and denominator helps determine horizontal or slant asymptotes and end behavior. Once these features are found, the graph can be sketched by combining algebraic structure with sign changes and a few test points.

Key Facts

A rational function has the form f(x) = P(x)/Q(x), where P and Q are polynomials and Q(x) != 0.
Vertical asymptotes occur at x = a when Q(a) = 0 and the factor causing Q(a) = 0 does not cancel.
A hole occurs when a common factor in P(x) and Q(x) cancels, such as f(x) = (x - 2)(x + 1)/(x - 2)(x - 3).
x-intercepts occur where P(x) = 0 after simplification, provided the point is not a hole.
If degree(P) < degree(Q), then the horizontal asymptote is y = 0. If degree(P) = degree(Q), then y = leading coefficient of P / leading coefficient of Q.
If degree(P) = degree(Q) + 1, the graph has a slant asymptote found by polynomial division.

Vocabulary

Rational function: A function written as one polynomial divided by another polynomial.
Vertical asymptote: A vertical line x = a that the graph approaches when the function values increase or decrease without bound near x = a.
Horizontal asymptote: A horizontal line y = b that the graph approaches as x becomes very large positive or negative.
Hole: A missing point on the graph caused by a factor that cancels from both the numerator and denominator.
Intercept: A point where the graph crosses or touches an axis, found from x = 0 for the y-intercept or y = 0 for x-intercepts.

Common Mistakes to Avoid

Canceling terms instead of factors, which is wrong because only common factors can be canceled in a rational expression. For example, x^2 - 4 and x - 2 must be factored before simplifying.
Using original numerator zeros as x-intercepts after cancellation, which is wrong because a canceled zero creates a hole, not an intercept. Always simplify first and then find intercepts.
Calling every denominator zero a vertical asymptote, which is wrong because some denominator zeros cancel and become holes. Check for common factors before deciding.
Finding the horizontal asymptote from intercepts or nearby points, which is wrong because horizontal asymptotes depend on end behavior and polynomial degrees. Compare the degrees and leading coefficients instead.

Practice Questions

1 For f(x) = (x^2 - 4)/(x - 2), simplify the function, state the location of any hole, and find the y-intercept.
2 For g(x) = (x + 3)/(x - 1), find the vertical asymptote, horizontal asymptote, x-intercept, and y-intercept.
3 A rational function has a factor (x - 5) in both the numerator and denominator, and after cancellation the simplified denominator still has a factor (x + 2). Explain why x = 5 is a hole and x = -2 is a vertical asymptote.