Three Types of Asymptotes

Algebra 2 · Axiom Academy

LESSON Three Types of Asymptotes Vertical, horizontal, and slant — the invisible guide lines a rational function approaches but never reaches. A rational function is undefined wherever its denominator is zero. If that zero doesn't cancel with the numerator, the function's values explode toward right beside it — and the graph shoots up (or down) along a vertical line it can never cross. At each uncancelled zero of the denominator Set the denominator to zero and solve: . The numerator ( 1 ) is not zero there, so the line survives. A horizontal asymptote describes the end behavior : the single height the graph flattens toward as . Which height (if any) depends only on how the degrees of the top and bottom compare. The bottom grows faster, so the graph flattens onto y = 0 . The graph levels off at the ratio of the leading coefficients. No horizontal asymptote — look for an oblique one instead. The top and bottom both have degree 2, so compare leading coefficients: the top's is 3 and the bottom's is 2 . When the top's degree is exactly one more than the bottom's, there is no horizontal asymptote — instead the graph settles onto a slanted line . Polynomial division splits the function into that line plus a remainder that vanishes as . The top has degree 2 and the bottom degree 1, and 2 = 1 + 1 — so an oblique asymptote exists. Divide: The quotient x + 3 is the asymptote; the remainder term shrinks to 0 as .

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