Graphing Rational Functions

Rational functions have lines the graph approaches but never reaches.

Vertical asymptotes: set the denominator to zero after cancelling. Any zero that cancelled with the numerator is a hole instead, not an asymptote.

Horizontal asymptotes come from comparing degrees:

• Bottom degree bigger → y = 0
• Degrees equal → y = ratio of the leading coefficients
• Top degree bigger by exactly 1 → no horizontal asymptote, but a slant one found by dividing

For y = (2x + 1)/(x − 3): vertical asymptote at x = 3, and since the degrees are equal, horizontal asymptote at y = 2/1 = 2.

A graph can cross a horizontal asymptote in the middle. It just cannot at the far ends.

For f(x) = P(x)/Q(x) in lowest terms, vertical asymptotes occur at the real zeros of Q; zeros common to P and Q that were cancelled produce removable discontinuities (holes) instead.

End behaviour compares degrees. With deg P < deg Q, the horizontal asymptote is y = 0. With deg P = deg Q, it is the ratio of leading coefficients. With deg P = deg Q + 1, polynomial division yields a linear quotient which is the slant (oblique) asymptote, with the remainder term vanishing at infinity. For larger degree gaps, the end behaviour follows the polynomial quotient and there is no linear asymptote.

Vertical asymptotes are never crossed, since the function is undefined there. Horizontal and slant asymptotes describe end behaviour only and may be crossed in the interior; solving f(x) = L finds any such crossings.

A complete sketch combines intercepts, asymptotes, holes and a sign analysis on the intervals between the zeros and the vertical asymptotes, which determines whether each branch approaches +∞ or −∞.