Graphs of Rational Functions | Precalculus (2026 Update) скачать в хорошем качестве

Graphs of Rational Functions | Precalculus (2026 Update)

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. 🔗 Visit www.clopensets.com for full class videos, guided notes, and printable worksheets that go along with this lesson. 🔗 ClopenSets shop: https://www.etsy.com/shop/Clopensets #education #maths #math #highschoolmath #highschool 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