Calculus 1 — 20.3: Asymptotes, Holes, and Intercepts

Описание: 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

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 3.6:  How to Sketch Gra...