AP Precalculus Cheat Sheet

The core ideas of AP Precalculus distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Function Transformations

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.

Polynomial Functions

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

Rational Functions

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

Exponential Growth and Decay

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.

Logarithms

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

Function Composition

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

Inverse Functions

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).

Trig and the Unit Circle

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.

Rates of Change

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.

Polar Coordinates

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

Piecewise Functions

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

Even and Odd

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

Key Terms at a Glance

Function:A relation that assigns exactly one output to each input in its domain. Passes the vertical line test when graphed.

Domain:The set of all valid input values (x-values) for a function. Restricted by division by zero, square roots of negatives, and logarithms of non-positives.

Range:The set of all output values (y-values) a function actually produces over its domain.

Transformation:An operation that modifies a parent function's graph through shifts, stretches, compressions, or reflections without changing its fundamental shape.

Parent Function:The simplest form of a function family, used as the starting point for transformations. Examples: $y = x^2$, $y = \sqrt{x}$, $y = |x|$, $y = e^x$.

Composition:The operation of applying one function to the output of another, written $(f \circ g)(x) = f(g(x))$. Evaluated from inside out.

Inverse Function:A function $f^{-1}$ that reverses $f$, satisfying $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Exists only for one-to-one functions.

One-to-One Function:A function where each output corresponds to exactly one input. Equivalently, it passes the horizontal line test. Only one-to-one functions have inverses.

Asymptote:A line that a function's graph approaches but never reaches (or reaches only in the limit). Can be vertical, horizontal, or oblique.

End Behavior:The trend of a function's output as x approaches positive or negative infinity. Determined by the leading term for polynomials and by asymptotes for rational and exponential functions.

Zero of a Function:An x-value where $f(x) = 0$. Also called a root or x-intercept. The multiplicity of a zero affects whether the graph crosses or bounces at that point.

Multiplicity:The number of times a factor $(x - r)$ appears in a polynomial's factored form. Determines graph behavior at the corresponding zero.

Exponential Function:A function of the form $f(x) = a \cdot b^x$ where $b > 0$ and $b \neq 1$. Characterized by a constant multiplicative rate of change over equal intervals.

Logarithmic Function:The inverse of an exponential function. $\log_b(x) = y$ means $b^y = x$. Domain is $x > 0$; range is all real numbers.

Unit Circle:A circle of radius 1 centered at the origin used to define trigonometric functions. The coordinates of a point at angle $\theta$ are $(\cos\theta, \sin\theta)$.

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