A polynomial is built from powers of x added together—like x³ − 4x. A rational function is a ratio of polynomials—like (x+1)/(x−2). Both show up in models of volume, cost, concentration, and more. The skill is reading the graph’s story: rises, falls, intercepts, and forbidden lines.
End behavior of polynomials
For large |x|, the highest-degree term dominates. An odd degree with positive leading coefficient goes down on the left, up on the right (like y = x³). Even degree with positive leading coefficient goes up on both ends (like y = x²).
Zeros: where the graph meets the x-axis
A zero is an input that makes the output zero—an x-intercept. Factoring often reveals them: (x−2)(x+1)=0 gives x = 2 and x = −1. Multiplicity (how many times a factor repeats) affects whether the graph crosses or bounces at that intercept.
Rational functions and asymptotes
Division by zero is not allowed. If the denominator is zero at x = a (and the numerator is not also zero there in a canceling way), the graph typically has a vertical asymptote—a vertical line it never crosses, approaching ±∞ nearby.
As |x| grows, many rationals approach a horizontal line (a horizontal asymptote) when the degrees of numerator and denominator allow it.
Common trap: Canceling a common factor can remove a hole or change a vertical asymptote into a removable discontinuity. Always check what was canceled before sketching.