Graphing Rational functions

Rational Function

A function of the form f(x) = p(x)/Q(x), where p and q are polynomials.

Hole

A point (x,y) at which there is a discontinuity in the graph, occurring when there is a common factor between the numerator and denominator.

Finding Holes

To find holes, set the common factor equal to 0 to find x-coordinates, then substitute back into the simplified function to find y-coordinates.

X-intercept(s)

The points where the graph crosses the x-axis, found by setting the numerator equal to 0 and solving.

Vertical Asymptote(s)

Vertical boundary lines where the graph will not cross, found by setting the denominator equal to 0 and solving.

Horizontal Asymptote

Horizontal guidelines that indicate the end behavior of the function, determined by comparing the degrees of the numerator and denominator.

Horizontal Asymptote Rules

If degree of p < degree of q, then y=0; if degree of p = degree of q, then y=c (the ratio of leading coefficients); if degree of p > degree of q, there is no horizontal asymptote.

Y-intercept

The point where the graph crosses the y-axis, found by substituting 0 for x in the simplified equation and solving for y.