MCV4U: Curve Sketching Rational Functions (Asymptotes, Holes & Oblique)

In Lesson 31, we apply our 10-step curve sketching algorithm to Rational Functions. Unlike polynomials, rational functions introduce new challenges like discontinuities, various types of asymptotes, and more complex derivatives.

In this lesson, we break down:

The Rational Function Checklist: Why checking the domain and finding discontinuities is your first priority.
Vertical & Horizontal Asymptotes: How to determine the behavior of the function as it approaches these boundaries.
Removable Discontinuities (Holes): Identifying when a factor cancels out and how to find the exact coordinates of the hole.
Oblique (Slant) Asymptotes: Using long division/simplification to find the linear path a function follows at its ends.
The First Derivative Challenge: Dealing with the quotient rule and algebraic manipulation to find critical points.
Case Studies:
Example 1: A standard rational function with two vertical asymptotes and a horizontal asymptote at $y=0$.
Example 2: A function with a hole at $(1, 2)$ and quadratic end behavior.
Example 3: A complex case featuring a vertical asymptote, a hole, and an oblique asymptote ($y = x + 2$).
Key Timestamps:

0:00 - Introduction: The Algorithm for Rational Functions
1:05 - Analyzing Asymptotes and End Behavior
3:45 - Example 1: $f(x) = \frac{x-4}{x^2-x-2}$ (Vertical & Horizontal Asymptotes)
7:05 - Finding Critical Points with the Product/Quotient Rule
11:00 - Using the Sign Table for Rational Derivatives
15:10 - Sketching the First Example Step-by-Step
19:15 - Example 2: $f(x) = \frac{x^3-x}{x-1}$ (Handling Removable Discontinuities)
22:30 - Derivative and Concavity for Example 2
26:40 - Example 3: $f(x) = \frac{x^3+x^2-x-1}{x^2-x}$ (The Oblique Asymptote Case)
32:10 - Finding and Sketching the Oblique Asymptote ($y=x+2$)
37:00 - Summary and Final Tips
Resources: Make sure to download the Rational Functions Practice Set from Google Classroom. Use Desmos to verify your sketches, but remember to "drag" over any holes to see the undefined points!

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