Complete analysis of a rational function — find all asymptotes, holes, intercepts, and domain, then graph it.
deg(num) = deg(den)+1 → oblique asymptote
Remember that asymptotes are guidelines, not walls — the function can sometimes cross a horizontal asymptote in the middle of the graph. The asymptote only describes the end behavior.
A vertical asymptote is different: the function never crosses it. The graph approaches but never reaches a vertical asymptote.
Degree rules for end behavior: If deg(num) < deg(den), HA is y = 0. If degrees are equal, HA is the ratio of leading coefficients. If deg(num) = deg(den) + 1, there is an oblique asymptote (use long division). If deg(num) > deg(den) + 1, there is no asymptote.
One-on-one Algebra 2 tutoring makes asymptotes, holes, and graphing click — we work through your actual problems and build the intuition that sticks for tests and beyond.