What is a vertical asymptote?

A vertical asymptote occurs at x-values where the denominator equals zero (and the numerator doesn't). For f(x) = 1/x, the denominator is zero at x = 0, so x = 0 is a vertical asymptote. Near this value, the function shoots to +∞ or -∞.

What is a horizontal asymptote?

A horizontal asymptote describes the behavior of the function as x → ±∞. For f(x) = 1/x, as x becomes very large, 1/x becomes very small (approaching 0). The curve gets closer and closer to y = 0 but never actually reaches it, so y = 0 is the horizontal asymptote.

Where are rational functions used in real life?

Rational functions model inverse relationships: speed and travel time (time = distance/speed), Boyle's law in chemistry (pressure × volume = constant, so P = k/V), electrical resistance in parallel circuits, and the intensity of light or sound decreasing with distance.

VIZMath · Function Guide

Rational Function

f(x) = 1/x

The hyperbola with asymptotes — models inverse relationships like speed/time and pressure/volume.

Rational Function

A rational function is a function that can be written as the ratio of two polynomials: f(x) = P(x)/Q(x). The simplest and most fundamental rational function is f(x) = 1/x, also called the reciprocal function.

Its graph is a hyperbola — two curved branches, one in the first quadrant (positive x and y) and one in the third quadrant (negative x and y). The function is undefined at x = 0, creating a vertical asymptote there.

As x grows very large, f(x) approaches but never reaches zero, creating a horizontal asymptote at y = 0. Rational functions are essential for modeling inverse relationships in science and engineering.

Standard Form

f(x) = a/(x - h) + k

Vertical stretch/compression — controls how quickly the curve approaches asymptotes

Horizontal shift — moves the vertical asymptote to x = h

Vertical shift — moves the horizontal asymptote to y = k

Key Properties

Domain

All x ≠ 0 (basic form); x ≠ h for transformed form

Range

All y ≠ 0 (basic form); y ≠ k for transformed form

Vertical Asymptote

x = 0 (the y-axis)

Horizontal Asymptote

y = 0 (the x-axis)

Odd Function

1/(-x) = -1/x — symmetric about the origin

Shape

Hyperbola — two branches in opposite quadrants

Examples

f(x) = 1/x

Basic reciprocal: hyperbola in quadrants 1 and 3.

f(x) = 2/x

Stretched vertically — branches curve further from origin.

f(x) = 1/(x - 2) + 1

Asymptotes shifted to x = 2 and y = 1.

Visualize the Rational Function

See the asymptotes and hyperbola branches shift as you transform the function

Try in VIZMath Pro →

FAQ

A rational function is any function written as one polynomial divided by another: f(x) = P(x)/Q(x). The simplest example is 1/x. The key features are asymptotes — lines the curve approaches but never crosses — and holes, which occur when numerator and denominator share a common factor.

Related Functions

Exponential Function Logarithmic Function