AdvancedProRational Functions
Rational Function
f(x) = P(x)/Q(x)
Overview
Rational functions are ratios of polynomials. Zeros of Q(x) create vertical asymptotes or holes.
Domain
All x where Q(x) ≠ 0
Range
Depends on function
Symmetry
Depends on function
Key Properties
1
Vertical asymptotes where Q(x) = 0
2
Holes where P and Q share a factor
3
Horizontal asymptote from degree comparison
4
Oblique asymptote if deg(P) = deg(Q)+1
Key Insight
Always factor both numerator and denominator first — common factors create holes, not asymptotes.
Visualize Rational Function
Interact with the graph, adjust parameters, and see how the function behaves in real time.