Rational Functions & Asymptotes

Calculus Readiness · Axiom Academy

A rational function is just one polynomial over another — and that single division is what makes its graph shoot to infinity, flatten out, and even skip a point. 1. Vertical Asymptotes — Where the Bottom Hits Zero A vertical asymptote is a vertical line x = a that the graph races toward but never touches, its value blowing up to or . It happens where the function tries to divide by zero. Watch : as x slides toward 1 , the denominator shrinks to nothing and the curve rockets up the dashed line . The denominator vanishes at x = 1 So x = 1 is a vertical asymptote Find the vertical asymptotes of . Factor the bottom: . Nothing cancels with x+1 , so set the denominator to zero: x = 2 and x = -2 . Two vertical asymptotes. 2. Holes — When a Factor Cancels Not every zero of the denominator is an asymptote. If that same factor also sits in the numerator, it cancels — and instead of blowing up, the graph just skips a single point . That missing point is a hole (a removable discontinuity). Watch collapse to the line y = x+1 , but with one point punched out. To locate the hole, plug the canceled value into the simplified function: at x = 2 , y = 2 + 1 = 3 . The hole sits at the point — drawn as an open circle. It does not cancel — the graph blows up. Vertical asymptote. The graph is otherwise smooth — one point is missing. Hole. 3. Horizontal Asymptotes — The Long-Run Level

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