Definition: A rational function is a ratio of two polynomials, f(x) = p(x)/q(x) with q(x) not zero, and its graph often has asymptotes and holes.
What is a rational function?
It is a fraction whose numerator and denominator are polynomials, for example f(x) = (2x + 6)/(x2 – 4). The domain excludes every x that makes the denominator zero.
What features does the graph have?
- A vertical asymptote where the denominator is zero and the numerator is not.
- A hole where a common factor cancels.
- A horizontal asymptote determined by comparing degrees.
- x-intercepts where the numerator is zero.
Worked example: f(x) = (2x + 6)/(x2 – 4) = (2x + 6)/((x – 2)(x + 2)). The domain excludes 2 and -2, where vertical asymptotes appear. The numerator is zero at x = -3, so the x-intercept is (-3, 0). The degree on top is smaller, so the horizontal asymptote is y = 0.
What is a common mistake?
Cancelling terms instead of factors. You can cancel (x – 2) from top and bottom, but you cannot cancel the x in (x + 5)/x. Mathbot says: factor first, then cancel.
Example: f(x) = (2x + 6)/(x^2 – 4) has asymptotes x = 2, x = -2, y = 0
Where to learn it
- Rational Functions: Asymptotes, Holes, and Graphs – lesson
- Polynomial and Rational Functions: Zeros and Asymptotes – lesson