Lesson 51: Graphing Rational Functions (Asymptotes, Holes, Intercepts)
Rational functions are ratios of polynomials. Their graphs have distinct features including vertical and horizontal asymptotes, holes, and intercepts. Understanding these features helps us sketch accurate graphs.
Steps to Graph Rational Functions
- Factor numerator and denominator: Simplify if possible.
- Identify vertical asymptotes: Values of x that make the denominator zero but not the numerator.
- Identify holes: Values that make both numerator and denominator zero.
- Find horizontal asymptotes: Compare degrees of numerator and denominator.
- Find x-intercepts: Set numerator equal to zero and solve.
- Find y-intercept: Evaluate function at x = 0.
- Sketch the graph: Plot asymptotes, intercepts, and draw curve respecting asymptotic behavior.
Examples
Example 1: f(x) = (x − 2)/(x² − 4)
- Factor: f(x) = (x − 2)/[(x − 2)(x + 2)] = 1/(x + 2), x ≠2
- Hole at x = 2
- Vertical asymptote: x = −2
- Horizontal asymptote: y = 0
- x-intercept: none (hole instead)
- y-intercept: f(0) = −1/2
Example 2: g(x) = (x² − 1)/(x − 1)
- Factor numerator: (x − 1)(x + 1)/(x − 1) → x + 1, x ≠1
- Hole at x = 1
- Vertical asymptotes: none (hole instead)
- Horizontal asymptote: none (linear function after simplification)
- x-intercept: x = −1
- y-intercept: g(0) = 1
Practice Problems
- Graph f(x) = (x + 3)/(x − 2)
- Graph g(x) = (x² − 9)/(x − 3)
- Graph h(x) = (x² − 4)/(x² − 1)
- Graph k(x) = (2x − 5)/(x² − 4x + 3)
- Graph m(x) = (x² + x − 6)/(x² − x − 6)