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Ap pre calculus Ration Functions
Precalculus • Functions
Lesson: Graphing Rational Functions
Learn how to sketch rational functions accurately by identifying intercepts, holes, vertical asymptotes, and end behavior asymptotes using a clear step-by-step method.
Lesson Objectives
- Understand what a rational function is
- Identify holes, intercepts, and asymptotes
- Determine end behavior using degree comparison
- Sketch rational functions with confidence
1. What Is a Rational Function?
A rational function is any function that can be written as:
f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
📷 Image placeholder: Examples of rational function graphs
2. Step-by-Step Graphing Method
- Factor the numerator and denominator completely
- Identify any holes from cancelled factors
- Find the y-intercept (x = 0)
- Find the x-intercepts (numerator = 0)
- Find vertical asymptotes (denominator = 0)
- Determine the end behavior asymptote
- Test behavior near vertical asymptotes
- Check whether the graph crosses the EBA
3. End Behavior Asymptotes (EBA)
Case 1: Degree of numerator < degree of denominator → y = 0
Case 2: Degrees equal → ratio of leading coefficients
Case 3: Degree of numerator > degree of denominator → slant or polynomial asymptote (use division)
📷 Image placeholder: Comparing the three EBA cases
4. Worked Example
Sketch the graph of:
y = (2x + 3)/(x − 2)
- No common factors → no holes
- y-intercept: (0, −3/2)
- x-intercept: (−3/2, 0)
- Vertical asymptote: x = 2
- Same degree → EBA: y = 2
- Graph does not cross the EBA
📷 Image placeholder: Final graph of the example
Practice Questions
- f(x) = (x + 1)/(x − 3)
- f(x) = (3x − 2)/(x + 4)
- f(x) = 5/(x − 1)
- f(x) = (x − 2)(x + 1)/(x − 2)(x − 3)
- f(x) = (2x + 5)/(x² + 1)
- f(x) = (4x² − 1)/(2x² + 7)
✅ Show Answer Key
Key ideas to check:
- Correct identification of VA and EBA
- Holes only when factors cancel
- Correct intercepts
- Clear asymptotic behavior
© Aviate Learning – Graphing Rational Functions