Asymptotes in Rational Functions

A rational function is the ratio of two polynomials, with a non-zero denominator.

A key feature of rational functions is their asymptotes—lines that the graph approaches but never touches.

These asymptotes can be vertical or horizontal.

Vertical asymptotes occur where the denominator equals zero, creating breaks in the graph. These are found by solving the denominator for zero.

Horizontal asymptotes describe the end behavior of a rational function as the input becomes very large or very small.

Rational functions may also cross horizontal asymptotes, unlike vertical ones, which act as strict boundaries.

The position of the horizontal asymptotes is determined by comparing the degrees of the numerator and denominator.

When the numerator’s degree is less than the denominator’s, the horizontal asymptote is y equals zero.

A greater degree in the numerator than the denominator means the function has no horizontal asymptote.

For instance, consider the concentration of a pollutant in water, modeled as a rational function. As water volume increases, the concentration of a pollutant decreases and approaches zero, forming a horizontal asymptote at y equals zero.