\[ f(x) = \frac{x^2-4}{x^2+x-6} \]
Concepts
- Holes (Removable Discontinuities): Occur at x-values that make BOTH the numerator and denominator zero (common factors).
- Vertical Asymptotes: Occur at x-values that make ONLY the denominator zero (after simplifying).
Click to factor the expression:
\[ \text{Expression Unfactored} \]
Common Factor: (x-2) → Hole at x=2
Remaining Denominator: (x+3) → VA at x=-3
Remaining Denominator: (x+3) → VA at x=-3
Rules
- Degree P < Degree Q: HA at \( y = 0 \) (x-axis).
- Degree P = Degree Q: HA at \( y = \frac{\text{Lead Coeff P}}{\text{Lead Coeff Q}} \).
- Degree P > Degree Q: No HA.
- Degree P = Degree Q + 1: Oblique (Slant) Asymptote. Find by Long Division.
Select an example:
\[ f(x) = \frac{3x^2-7x}{x^2-x-6} \]
Degrees are equal (2 = 2) → HA: y = 3/1 = 3
\[ f(x) = \frac{x^2-x-2}{x+3} \]
Finding Intercepts
- y-intercept: Find \( f(0) \). (Plug in 0 for x).
- x-intercepts: Find zeros of the Numerator (where \( P(x) = 0 \)) after simplifying.