1.2.1—Rational graphs

For a rational function, factorise numerator and denominator first. A denominator zero may create a vertical asymptote or a removable hole; the degrees and leading coefficients determine horizontal or oblique behaviour.

Find excluded x-values, intercepts and the sign of each factor on intervals. Long division gives the polynomial asymptote when the numerator degree is at least the denominator degree.

For (x+1)/(x−2), x=2 is a vertical asymptote, x=−1 is the x-intercept, and y=1 is the horizontal asymptote because the leading coefficients match.

A cancelled factor creates a hole rather than a vertical asymptote, and an asymptote is a limiting feature—not necessarily a line the graph can never cross in every case.