Rational Function Analyzer
Analyze a rational function of the form (num_a·x + num_b) / (den_a·x + den_b) to find its vertical asymptote, horizontal asymptote, domain, and intercepts. Useful for precalculus and calculus students.
About this calculator
A rational function has the form R(x) = (num_a·x + num_b) / (den_a·x + den_b). Its vertical asymptote occurs where the denominator equals zero: x = −den_b / den_a. The horizontal asymptote is determined by the ratio of leading coefficients: y = num_a / den_a (when degrees are equal). The x-intercept is found by setting the numerator to zero: x = −num_b / num_a. The y-intercept is R(0) = num_b / den_b. The domain excludes any x value that makes the denominator zero. Together these features give a complete picture of the function's behavior, including end behavior and any discontinuities.
How to use
Analyze R(x) = (2x + 6) / (3x − 9). Enter num_a = 2, num_b = 6, den_a = 3, den_b = −9. Vertical asymptote: x = −(−9) / 3 = 3. Horizontal asymptote: y = 2 / 3. X-intercept: x = −6 / 2 = −3, so the graph crosses the x-axis at (−3, 0). Y-intercept: R(0) = 6 / (−9) = −2/3, so the graph crosses the y-axis at (0, −2/3). Domain: all real numbers except x = 3.
Frequently asked questions
What is the difference between a vertical and a horizontal asymptote in a rational function?
A vertical asymptote is a vertical line x = c where the function grows without bound because the denominator equals zero at c. A horizontal asymptote is a horizontal line y = L that the function approaches as x → ±∞, determined by the ratio of leading coefficients when the numerator and denominator have the same degree. The function can never cross a vertical asymptote (within its domain), but it can cross a horizontal asymptote at finite x values. Both asymptotes are essential for sketching the graph accurately.
How do you find the domain of a rational function?
The domain of a rational function is all real numbers except those that make the denominator zero. To find these excluded values, set the denominator polynomial equal to zero and solve. For (num_a·x + num_b) / (den_a·x + den_b), this means den_a·x + den_b = 0, giving x = −den_b / den_a. If the denominator has higher-degree factors, each real root is excluded from the domain. In interval notation, you express the domain as the real line with those points removed.
When does a rational function have a hole instead of a vertical asymptote?
A hole (removable discontinuity) occurs when a factor cancels between the numerator and denominator. For example, (x² − 1) / (x − 1) has a factor (x − 1) in both numerator and denominator; after cancellation it simplifies to (x + 1), with a hole at x = 1 rather than a vertical asymptote. In contrast, a vertical asymptote occurs when the denominator factor does not cancel. This calculator focuses on linear numerators and denominators, so holes arise only if num_a / num_b = den_a / den_b (proportional coefficients).