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Rational Functions & Asymptotes
AP Precalculus • Unit 1
Rational Functions & Asymptotes
Learn how rational functions behave near asymptotes, how to identify holes and vertical asymptotes, and how algebra determines graph shape.
Lesson Objectives
- Define rational functions
- Identify vertical and horizontal asymptotes
- Recognize holes in graphs
- Sketch rational function graphs conceptually
1. What Is a Rational Function?
A rational function is a function that can be written as a ratio of two polynomials:
f(x) = P(x) / Q(x), where Q(x) ≠ 0
Examples:
f(x) = (x + 1)/(x − 2)
g(x) = (2x² − 3)/(x + 5)
f(x) = (x + 1)/(x − 2)
g(x) = (2x² − 3)/(x + 5)
📌 Diagram placeholder: basic rational function graph
2. Vertical Asymptotes
A vertical asymptote occurs where the denominator equals zero and does not cancel.
Example (from the book):
f(x) = 1/(x − 3)
f(x) = 1/(x − 3)
- Denominator = 0 when x = 3
- Vertical asymptote at x = 3
Important:
If a factor cancels, the graph has a hole, not an asymptote.
If a factor cancels, the graph has a hole, not an asymptote.
📊 Diagram placeholder: vertical asymptote
3. Holes in Rational Functions
A hole occurs when a factor cancels from both the numerator and denominator.
Example (OpenStax style):
f(x) = (x − 2)(x + 1)/(x − 2)
f(x) = (x − 2)(x + 1)/(x − 2)
- Factor (x − 2) cancels
- Hole at x = 2
After simplification: f(x) = x + 1 (but with x ≠ 2)
📌 Diagram placeholder: hole in rational graph
4. Horizontal Asymptotes
Horizontal asymptotes describe the behavior of the graph as x → ±∞.
Rules (from the book):
- If degree of numerator < degree of denominator → y = 0
- If degrees are equal → y = ratio of leading coefficients
- If numerator degree > denominator → no horizontal asymptote
Example:
f(x) = (2x + 1)/(x − 4) → horizontal asymptote y = 2
f(x) = (2x + 1)/(x − 4) → horizontal asymptote y = 2
📈 Diagram placeholder: horizontal asymptote comparison
5. Summary Table: Features of Rational Functions
| Feature | How to Find |
|---|---|
| Vertical Asymptote | Denominator = 0 (no cancellation) |
| Hole | Common factor cancels |
| Horizontal Asymptote | Compare degrees |
Practice Questions
- Find the vertical asymptote of f(x) = 3/(x + 1)
- Does f(x) = (x − 4)/(x − 4) have a hole or asymptote?
- Find the horizontal asymptote of f(x) = (5x − 1)/(2x + 3)
✅ Show Answer Key
- x = −1
- Hole at x = 4
- y = 5/2
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