Polynomial and rational functions

Polynomials: sums of powers

A polynomial combines powers of x with coefficients—like x³ − 4x or 2x² + 3x − 1. They are the smooth, unbroken curves of precalculus: no sudden vertical walls. Degree (highest power) and the leading coefficient control the big-picture shape at the far left and right—called end behavior.

• Odd degree, positive leading coefficient: down on the left, up on the right (like y = x³).
• Even degree, positive leading coefficient: up on both ends (like y = x²).

Zeros are inputs that make the output zero—x-intercepts. Factoring reveals many of them. Multiplicity (repeated factors) decides whether the graph crosses or bounces at an intercept.

Where polynomials appear

• Area and volume formulas expanded in a variable side length
• Simple physics models (position under constant acceleration involves t²)
• Economics cost/revenue sketches in intro models
• Design curves approximated by polynomial pieces in graphics

Rational functions: ratios

A rational function is a quotient of polynomials, such as (x+1)/(x−2). Division by zero is forbidden, so values of x that zero the denominator (without a matching canceling factor) typically create a vertical asymptote—a vertical line the graph approaches but does not cross, shooting toward ±∞.

As |x| grows, many rationals settle toward a horizontal line (a horizontal asymptote), which describes long-run behavior—like a concentration approaching a limit.

Everyday rational intuition

• Sharing a fixed amount — pizza slices per person ≈ total / number of people (a 1/x shape).
• Speed = distance/time — for fixed distance, time in the denominator.
• Unit prices — cost per item when packing more into a bulk deal.
• Density — mass/volume style ratios in science.