AP Precalculus: Study Guide & Free Lessons
Short lessons, worked examples, and practice that connects.
Function change and transformations
A function pairs each allowed input with one output. Average rate of change compares two outputs over a stated input interval.
Polynomial structure and end behavior
A polynomial combines nonnegative integer powers. Its leading term controls end behavior, while all terms can matter near a particular input.
Zeros, factors and remainders
A zero r of a polynomial corresponds to a factor x-r. A remainder on division by x-r equals the value of the polynomial at r.
Rational functions and restrictions
A rational function is a quotient of polynomials. Inputs making the original denominator zero remain excluded even after cancellation.
Exponential growth and decay
Equal input increments produce equal multiplicative output factors in an exponential model. The initial value and growth factor play different roles.
Logarithms as inverse questions
The logarithm log_b(y) asks which exponent on b produces y, for b positive and not 1 and for y positive.
Compose and reverse functions
Composition applies the inner rule first. An inverse reverses a one-to-one mapping on a specified domain.
Angles and the unit circle
Radian measure relates an arc length to its radius. On the unit circle, cosine is the horizontal coordinate and sine the vertical coordinate.
Read a sinusoidal model
A sinusoidal model describes repeating change using a midline, amplitude, period and horizontal position.
Polar coordinates and equivalent points
Polar coordinates locate a point using a directed distance and angle. Different coordinate pairs can represent the same point.
Sequences as discrete models
A sequence is a function with a discrete input, often an index. Arithmetic and geometric sequences encode additive and multiplicative patterns.
Select and interpret a model
A model is a mathematical description whose assumptions must match the data and the question. Fit alone does not establish a causal explanation.
Reconstruct real polynomials from complex zeros
Use conjugate pairing and multiplicity to recover a real polynomial.
Use multiplicity and signs without expanding
Infer crossings, sign intervals and end behavior from a factored polynomial.
Separate holes from vertical and slant asymptotes
Classify different failures and long-run features of a rational rule.
Solve rational equations with admissible inputs
Distinguish an algebraic candidate from a solution in the original domain.
Fit and compare exponential models from separated observations
Recover a per-unit factor and test an alternative model using an extra observation.
Solve logarithmic equations and inequalities with domains
Use logarithm rules without retaining forbidden or extraneous candidates.
Read multiplicative change on a semi-log display
Translate straight-line coordinates in log space into an exponential rule.
Enumerate trigonometric solutions on a stated interval
Find all branches and respect included and excluded interval endpoints.
Respect the principal range of an inverse trigonometric function
Distinguish inverse evaluation from unrestricted equation solving.
Use reciprocal identities with quadrant signs
Recover an unknown trigonometric value while keeping its sign.
Fit phase and frequency from observed peaks
Construct a sinusoid from extrema and timing, then interpret changes in context.
Connect polar curves, directed radii and rates
Translate polar geometry and compare radial changes over angle intervals.
Expanded foundation practice, not complete AP coverage or a calibrated AP score estimate.
A clear route through the material
Follow the topics in order or use the prerequisites to choose a starting point. Read the worked example before focused practice, then use review cards to revisit the idea.
Continue your preparation
Scope: Function reasoning foundations