Introduction to Rational Functions
A rational function is a function that can be expressed as the quotient of two polynomials:
Where P(x) and Q(x) are polynomials and Q(x) ≠ 0.
Rational functions often have asymptotes and discontinuities that define their behavior. Understanding these elements is crucial for accurate graphing.
Vertical Asymptotes
Vertical asymptotes occur where the denominator equals zero (and the numerator is not zero at the same point).
Method: Set the denominator equal to zero and solve for x.
Example: For f(x) = (x + 2)/(x² - 4), we have x² - 4 = 0, so x = ±2. These are vertical asymptotes.
Example 1: Finding Vertical Asymptotes
Function: f(x) = (x + 1)/(x² - 9)
Set denominator to zero: x² - 9 = 0
Solution: x = ±3
Vertical asymptotes at x = -3 and x = 3
Horizontal Asymptotes
Horizontal asymptotes describe the behavior of the function as x approaches ±∞.
Rules:
- If degree of numerator < degree of denominator: Horizontal asymptote at y = 0
- If degree of numerator = degree of denominator: Horizontal asymptote at y = a/b (ratio of leading coefficients)
- If degree of numerator > degree of denominator: No horizontal asymptote (may have slant asymptote)
Example 2: Finding Horizontal Asymptotes
Function: f(x) = (2x² + 3)/(x² - 1)
Both numerator and denominator have degree 2
Horizontal asymptote at y = 2/1 = 2
Intercepts
X-Intercepts
Set f(x) = 0 and solve for x. This occurs when the numerator equals zero.
Y-Intercept
Find f(0) by substituting x = 0 into the function.
Example 3: Finding Intercepts
Function: f(x) = (x² - 4)/(x + 1)
X-intercepts: x² - 4 = 0 → x = ±2
Y-intercept: f(0) = (0² - 4)/(0 + 1) = -4
Discontinuities
Discontinuities occur where the function is undefined:
- Vertical asymptotes: Where denominator = 0 and numerator ≠ 0
- Holes: Where both numerator and denominator equal zero (common factors)
Special Case: Holes in the Graph
If a factor (x - a) appears in both numerator and denominator, there is a hole at x = a.
Example: f(x) = (x² - 4)/(x - 2) = (x - 2)(x + 2)/(x - 2)
There is a hole at x = 2 (not an asymptote).
Approaching Asymptotes
To determine if the function approaches an asymptote from above or below:
- Choose test points near the asymptote
- Evaluate f(x) for values slightly less than and greater than the asymptote
- Observe the sign of f(x) to determine approach direction
Example 4: Behavior Near Asymptote
Function: f(x) = 1/(x - 2)
As x → 2⁻ (approaching from left): f(x) → -∞
As x → 2⁺ (approaching from right): f(x) → +∞
Quadratic Factorization
Factorizing quadratics is essential for finding zeros and simplifying rational functions.
For ax² + bx + c = 0, we look for two numbers that multiply to ac and add to b.
This is the quadratic formula, used when factoring is difficult or impossible.
Example 5: Factorization
Factor: x² - 5x + 6
Find two numbers that multiply to 6 and add to -5: -2 and -3
Factorization: (x - 2)(x - 3)