AP Precalculus Study Guide
Four units following the College Board framework. Units 1 through 3 are assessed on the exam; Unit 4 is taught but never tested. Each unit explains what to know, how to apply it, what typically goes wrong, and one study move that makes the material stick.
Four units
Polynomial and Rational Functions; Exponential and Logarithmic Functions; Trigonometric and Polar Functions; then Functions Involving Parameters, Vectors, and Matrices.
Three are assessed
The AP exam covers Units 1 through 3 only. Unit 4 is part of the course but never appears on the exam.
Grade levels
Usually taken in grades 10 through 12, after Algebra 2, as preparation for AP Calculus or AP Statistics.
Graphing calculator
Required. Part of the exam assumes one, and several question types are written to be impractical without it.
Units
Unit 1 is the largest by a wide margin. The rest move faster, and the last one is the bridge into calculus.
Unit 1 · Largest assessed unit
The unit that establishes the language for the rest of the course. It introduces rate of change as the thing that distinguishes function families, then works through polynomial behavior and the more delicate business of rational functions, where the graph depends on structure that is invisible until you factor.
Unit 2 · Moderate
A short unit with a single organising idea: exponential functions change by equal ratios over equal input intervals, and logarithms are the inverse that recovers the exponent. Most of the unit is consequences of that sentence, including modeling, equation solving, and the semi-log plot.
Unit 3 · Large assessed unit
The unit that introduces periodicity. It builds the unit circle, turns it into sinusoidal functions with four adjustable parameters, uses those to model repeating phenomena, and then changes coordinate systems entirely to look at polar functions.
Unit 4 · Not assessed on the exam
Taught as part of the course but never tested. It covers parametric and implicitly defined curves, vectors, and matrices as transformations. Worth learning for later courses in calculus and linear algebra, but it should not consume revision time in the weeks before the exam.
Mathematical Practices
The three AP mathematical practices run through every unit rather than being taught once and set aside. These are the habits they amount to when a student is actually answering a question.
Solve, manipulate, and rewrite without error. This is the floor rather than the goal: an exam question rarely stops at the computation, but a computation error ends the question regardless of how good the reasoning was.
Move between equation, graph, table, and verbal description, and treat each as evidence about the same object. Many AP Precalculus items give you one representation and ask a question that is far easier in another.
State a claim, name the evidence, and connect the two. On free-response work the difference between full and partial credit is usually the sentence explaining why the chosen function family or parameter fits, not the number itself.
Give every parameter its meaning and units. A model is not finished when the equation is written; it is finished when you can say what the base, the coefficient, and the shift mean for the situation being described.
Assessment
Two lists: one for selected-response work, one for anything you have to write out.
Study plan
Four moves, in the order that pays off.
Average rate of change over an interval, and what constant first or second differences imply about degree, run through the whole course. Practise reading it from a table and from a graph, not only from a formula.
Unit 1 turns on one habit: factor both parts of a rational function before saying anything about its graph. Build it early and Unit 1 becomes mechanical.
An exponential is the family whose outputs change by a constant factor over equal input intervals. Anchor Unit 2 on that sentence and the growth factor, the decay factor, and the semi-log plot all follow from it.
Learn the special triangles and the symmetry of the circle rather than a table of values. A memorised table fails on unfamiliar angles; the structure does not.
For every model you build, say what each parameter means in the situation. This is the habit that separates a mid-range free-response score from a high one, and it cannot be learned the night before.
Know which operations your calculator does quickly and which it does slowly. Deciding mid-exam how to run a regression is expensive.
Curriculum scope
This is the College Board AP Precalculus framework, not a generic precalculus syllabus. The organising idea across all four units is that a function family is identified by how its output changes, and that a claim about a function must be justified from an equation, a graph, a table, or a context rather than asserted.
Essential foundations
Included in this sequence
Compare with the school syllabus
Glossary
Average rate of change
The change in output divided by the change in input across an interval, equal to the slope of the line joining the endpoints.
Multiplicity
The exponent on a factor at a zero, which decides whether the graph crosses the axis or touches and turns.
End behavior
What a function does as the input grows without bound in either direction, determined by the leading term alone.
Hole
A single missing point on a rational graph, produced by a factor that cancels from both numerator and denominator.
Vertical asymptote
A line the graph approaches without meeting, produced by a denominator factor that does not cancel.
Growth factor
The constant an exponential model multiplies by over one period, equal to one plus the growth rate.
Semi-log plot
A plot with a logarithmic vertical axis, on which exponential data appears as a straight line.
Midline
The horizontal line halfway between the maximum and minimum of a sinusoidal function.
Amplitude
Half the distance between the maximum and the minimum of a sinusoidal function.
Phase shift
The horizontal translation of a sinusoid, read only after the coefficient of the variable has been factored out.
Pole
The origin of a polar coordinate system, from which the radius is measured.
Parameter
A third variable that generates the coordinates of a curve and records how the curve is traversed.
Determinant
A number computed from a square matrix that is zero exactly when the matrix is not invertible.
