Calculus 1 — 20.3: Asymptotes, Holes, and Intercepts скачать в хорошем качестве

Calculus 1 — 20.3: Asymptotes, Holes, and Intercepts

A six-step checklist for analyzing any rational function — from factoring to graphing. Learn how to distinguish holes from vertical asymptotes, apply the degree-comparison shortcut for horizontal asymptotes, and find intercepts by checking domain restrictions. Two fully worked examples and a challenge problem show how the same systematic process handles every case. Key concepts covered: • Rational functions as ratios of polynomials • The six-step analysis checklist: factor, cancel, classify, compare degrees, find intercepts • Holes (removable discontinuities) vs. vertical asymptotes (non-removable discontinuities) • Why canceling common factors reveals holes, not asymptotes • Finding the y-coordinate of a hole using the simplified function • Horizontal asymptote rules by degree comparison: numerator degree less than, equal to, or greater than denominator degree • Leading coefficient ratio when degrees are equal • Why functions can cross horizontal asymptotes in the interior of the domain • X-intercepts from numerator zeros and y-intercepts from f(0) • Why a vertical asymptote at x = 0 prevents a y-intercept from existing • Worked Example 1: f(x) = (x² − 1)/x³ — no holes, VA at x = 0, HA at y = 0, no y-intercept • Worked Example 2: f(x) = (2x² − 8)/(x² − 16) — two vertical asymptotes, HA at y = 2, curve crossing the HA • Challenge Problem: f(x) = (x² − 4x + 3)/(x² − 1) — identifying a hole at (1, −1) vs. a vertical asymptote at x = −1 ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 3.6:  How to Sketch Gra...