Rucete ✏ AP Precalculus In a Nutshell
3. Rational Functions
This chapter explores rational functions—their graphs, discontinuities, asymptotes, and how to solve rational inequalities. You'll learn to use factoring, division, and analysis to interpret complex rational expressions both algebraically and graphically.
- What Is a Rational Function?
- A rational function is the quotient of two polynomials: f(x) = P(x)/Q(x), where Q(x) ≠ 0.
- May include discontinuities:
- Vertical asymptotes: Q(x) = 0, P(x) ≠ 0
- Holes (removable): same zero in numerator and denominator
- Key features: Intercepts, asymptotes, holes, end behavior
- Long Division of Rational Expressions
- Used to rewrite rational functions in equivalent forms
- Helps identify end behavior and simplify expressions
- Polynomial long division:
- f(x)/g(x) = q(x) + r(x)/g(x), where r(x) has lower degree than g(x)
- Example: (2x³ + 5x² − 3x + 7)/(x² + 6) → 2x + 5 + (−15x − 23)/(x² + 6)
- Intercepts and Zeros
- x-intercepts: Set numerator = 0 and ensure denominator ≠ 0
- y-intercept: f(0), if defined
- Zeros = where graph crosses the x-axis
- Vertical Asymptotes
- Occur where denominator = 0, but numerator ≠ 0
- Notation:
- limₓ→a⁻ f(x) = ±∞
- limₓ→a⁺ f(x) = ±∞
- Graph gets closer to vertical line but never crosses it
- Holes (Removable Discontinuities)
- Occur when numerator and denominator share a factor
- Steps:
- Factor both numerator and denominator
- Cancel common factor
- Plug in x-value of hole into simplified function to find y
- Graph hole with open circle at (x, y)
- End Behavior Asymptotes
- Determined by degree comparison between numerator and denominator:
- Degree numerator < denominator → y = 0
- Degrees equal → y = leading coeff. ratio
- Degree numerator > denominator → slant or curved asymptote
- Use long division to find slant/curved asymptotes
- Seven-Step Strategy to Graph Rational Functions
- Factor numerator and denominator
- Find x- and y-intercepts
- Determine end behavior asymptote
- Find vertical asymptotes
- Check for symmetry using f(−x)
- Test a point in each region divided by asymptotes
- Sketch the graph using all features
- Symmetry
- f(x) = f(−x) → even function → symmetric about y-axis
- f(x) = −f(−x) → odd function → symmetric about origin
- If neither, function is not symmetric
- Rational Inequalities
- Solving steps:
- Rewrite as a single fraction
- Find critical values (zeros & vertical asymptotes)
- Plot critical points on number line
- Use test points in intervals
- Choose intervals that satisfy the inequality
- Use open/closed circles on number lines to show whether endpoints are included
In a Nutshell
Rational functions combine polynomial expressions into powerful and complex curves. Their graphs can have asymptotes, intercepts, and holes. Understanding these features and how to analyze them through factoring, long division, and strategic graphing lays the foundation for solving rational equations and inequalities confidently.