AP Precalculus

Moving, stretching, or flipping a graph. Changes to the input (inside) move left/right. Changes to the output (outside) move up/down. That inside/outside split trips people up—but it's the key.

Functions built from powers of x. The highest power tells you how the ends behave: odd degree = opposite ends; even degree = same ends.

One polynomial divided by another. Vertical asymptotes happen where the bottom is zero. Holes happen when top and bottom share a factor.

Multiply by the same factor over equal steps. Growth when the base > 1; decay when it's between 0 and 1. Used for populations, money, radioactivity.

Logs undo exponents. $\log_b(y)=x$ means "to what power do we raise b to get y?" Turns multiplication into addition.

Plug one function into another. Do the inner one first; use that result as the input to the outer one. Order matters.

A function that undoes another. $f^{-1}$ takes the output and gives back the input. Only works for one-to-one functions (each output comes from exactly one input).

Sine and cosine come from a circle of radius 1. cosine = x-coordinate, sine = y-coordinate. They repeat every $2\pi$—perfect for waves, tides, anything that cycles.

How much the output changes per unit of input. For a line it's constant (the slope). For curves it varies—that tells you if the graph is curving up or down.

Location by distance (r) and angle (θ) instead of x and y. Great for spirals, petals, anything that spins.

Different rules for different x-values. Evaluate by asking which rule applies to your input.

Even: flipping left-right leaves the graph the same ($x^2$, cos). Odd: flipping gives the opposite ($x^3$, sin).