Rational Functions, Explained Visually — Asymptotes, Holes & End Behavior

Everything a rational function's graph can do, one feature at a time. It starts with a real problem — the cheapest tin can — that produces a rational cost function, then works through the whole toolkit: reduce to lowest terms, find the vertical asymptotes and the holes, and settle the ends with the degree rules for horizontal and oblique asymptotes. A full worked example does an oblique asymptote by long division, and the lesson closes by reading a function backwards from its graph. No calculus needed. ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ WHAT YOU'LL LEARN ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ • The definition of a rational function, and its domain • Reducing a rational function to lowest terms • Vertical asymptotes — found from the reduced denominator • Multiplicity at a vertical asymptote: whether the graph flips sign across it • Holes (removable discontinuities), and how they differ from asymptotes — with the hole's coordinates • Horizontal vs. oblique (slant) asymptotes, and the degree rules that decide which you get • A full worked oblique-asymptote example by polynomial long division • Why a rational function can never have both a horizontal and an oblique asymptote • Reading a rational function's formula from its graph ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ CHAPTERS ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ 0:00 The cost-of-a-can problem 2:05 What a rational function is 3:24 Reducing to lowest terms 6:52 Vertical asymptotes: the picture first 8:01 Multiplicity at a vertical asymptote 9:12 Vertical asymptotes: two examples 10:54 Holes 12:27 Horizontal & oblique asymptotes: definitions 13:37 Which end asymptote? The degree rules 14:44 Horizontal-asymptote examples 15:28 Oblique asymptote: a worked example 18:29 Reading a graph backwards 19:41 Capstone check 20:41 Recap 21:27 Check yourself (quiz) ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ ABOUT ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━ Part of the Applied Calculus series.