Graphs of Rational Functions | Precalculus (2026 Update)

Описание: In Lesson 2.7, we focus on graphing rational functions by hand using a clear step-by-step checklist. You’ll learn how to find horizontal asymptotes from powers, factor to check for holes, find vertical asymptotes from the denominator, and then locate intercepts to build a strong sketch. We work several examples, including cases with no x-intercepts, graphs with multiple vertical asymptotes, and a rational function that includes a removable discontinuity (hole). We finish with a quick introduction to slant (oblique) asymptotes and when they occur.

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Chapters
00:00 Intro + what you should learn (Lesson 2.7)
00:19 Graphing checklist for rational functions (asymptotes, holes, intercepts, then finish with tech/table)
01:11 Picking test points by region to finish the sketch
02:01 Example 1: sketching using asymptotes (including using the parent function idea)
03:48 Example 2: asymptotes + intercepts (including when there is no x-intercept)
05:35 Example 3: ratio horizontal asymptote + vertical asymptote at x = 0 + intercept behavior
07:46 Example 4: multiple vertical asymptotes (factoring denominator) + using test points/technology
10:16 Example 5: holes (cancellation), new function, intercepts, and final sketch
12:26 Slant (oblique) asymptotes: when they happen + long division idea