MathIsimple

Rational Functions & Asymptotes

Master rational function properties, understand vertical and horizontal asymptotes, identify holes, and learn comprehensive graphing techniques.

11th Grade
Functions
~60 min
🎮 Interactive Activity: Vertical Asymptote Finder

Find the vertical asymptote(s) of the rational function!

Rational Function:

f(x) = (x + 2)/(x - 3)
🎮 Interactive Activity: Horizontal Asymptote Predictor

Predict the horizontal asymptote based on degrees!

Rational Function:

f(x) = (degree 2) / (degree 3)
↓

Horizontal Asymptote:

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1. Rational Function Definition

What is a Rational Function?

A rational function is a function that can be expressed as the ratio of two polynomials:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

Key characteristics:

  • Domain excludes values where Q(x) = 0
  • May have vertical asymptotes at zeros of Q(x)
  • May have horizontal or oblique asymptotes
  • May have holes where factors cancel
Example 1: Simple Rational Function

f(x) = 1/(x - 2)

• Numerator: 1 (constant polynomial)

• Denominator: x - 2 (linear polynomial)

• Domain: All real numbers except x = 2

• Vertical asymptote: x = 2

• Horizontal asymptote: y = 0

Example 2: More Complex Rational Function

f(x) = (x² - 4)/(x² - x - 2)

• Numerator: x² - 4 = (x - 2)(x + 2)

• Denominator: x² - x - 2 = (x - 2)(x + 1)

• After simplifying: f(x) = (x + 2)/(x + 1) for x ≠ 2

• Hole at x = 2, vertical asymptote at x = -1

2. Domain of Rational Functions
3. Vertical Asymptotes
4. Horizontal Asymptotes
5. Holes (Removable Discontinuities)
6. Oblique (Slant) Asymptotes
7. Graphing Rational Functions
8. Real-World Applications
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
0
Correct
0%
Score
1
What is the vertical asymptote of f(x) = (x + 3)/(x - 2)?
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2
What is the horizontal asymptote of f(x) = (3x² + 1)/(2x² - 5)?
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3
For f(x) = (x + 1)/(x² - 4), what are the vertical asymptotes?
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4
What is the horizontal asymptote of f(x) = (2x + 1)/(x² - 3)?
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5
If f(x) = (x² - 1)/(x - 1), what happens at x = 1?
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6
What is the horizontal asymptote of f(x) = (4x³ - 2x)/(2x³ + 1)?
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7
For f(x) = (x + 2)/(x² + 1), how many vertical asymptotes are there?
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8
What is the domain of f(x) = (x - 3)/(x² - 9)?
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9
If the numerator degree is greater than the denominator degree, what type of asymptote exists?
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10
What is the y-intercept of f(x) = (2x - 4)/(x + 1)?
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