Rational Functions, Asymptotes and Curve Sketching
Sketch a quotient of polynomials by finding its zeros, its vertical asymptotes and what it does far from the origin.
What a learner can do afterwards
- Locate vertical asymptotes from the denominator's zeros and check whether a factor cancels into a hole
- Decide the behaviour for large values by comparing the degrees of numerator and denominator
- Combine intercepts, asymptotes and sign changes into a sketch and check it against a few points
1 · Read
A rational function is one polynomial divided by another, and the bottom one runs the show. Factor top and bottom fully first. If a factor cancels on both levels, the break is a removable hole, a single missing point. Otherwise each surviving zero of the denominator gives a vertical asymptote, where the graph shoots toward infinity. Zeros of the surviving top give the x-intercepts.
Far from the origin only the biggest terms matter, so compare degrees to predict the end behavior. A heavier bottom drags the graph to zero. Equal degrees settle at the ratio of the leading coefficients. A heavier top grows without bound. Near each break, test the sign on both sides to learn whether branches dive or climb, then check a plotted point in every region.
Work three cases. For (x + 1) / (x - 2), nothing cancels, so x = 2 is a vertical asymptote. For (x - 3)(x + 1) over (x - 3)(x - 2), the shared factor cancels, leaving a hole at x = 3 with an asymptote at x = 2. For (3x squared + 1) / (2x squared - 5), degrees match, so the graph settles at y = 3/2.
Build the habit in order: factor first, cancel second, solve last. Skipping the middle step turns holes into fake asymptotes. Finish every sketch with a sign check plus one plotted point per region.
Factor, cancel holes, solve for asymptotes, then read end behavior off the degrees.
2 · Watch
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Where it sits
Learn first
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Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.