AP Precalculus: what it covers (the four units in our words), how it maps to MHF4U and a precalculus course, the modelling habit, and how to study for the exam

AP Precalculus is the College Board's course on the functions calculus assumes: polynomial and rational functions, exponential and logarithmic functions, trigonometric and polar functions, and a fourth unit on parameters, vectors and matrices that the exam does not test. Its habit is modelling: every function is treated as a description of something changing, with rates of change described in words. It overlaps most of Ontario's MHF4U and an ordinary precalculus course. Here is what it covers, where it meets those courses, one worked example per examined unit, and a plan.

What AP Precalculus is, and the four units in our words

AP Precalculus is a newer AP course. Schools first taught it in the 2023–24 school year, and its first exam was in May 2024. Because it is new, the College Board still revises its course and exam description; read the current one before you plan.

The four AP Precalculus units, described in our words:

1. Polynomial and rational functions. End behaviour, zeros and their multiplicity, rates of change, and rational functions with their asymptotes and holes.
2. Exponential and logarithmic functions. Sequences that grow by adding or by multiplying, exponential and logarithmic functions as inverses, the laws of logarithms, and models fitted to data.
3. Trigonometric and polar functions. Sine, cosine and tangent as periodic functions, their transformations and inverses, equations and identities, and polar coordinates and polar graphs.
4. Functions involving parameters, vectors and matrices. Parametric functions, vectors, and matrices as transformations.

The exam assesses units 1 to 3. Unit 4 is part of the course framework, taught at the school's choice, and not examined.

The course's habit, in one sentence: functions are models of change, so you are asked to describe rates of change in words and to read them from tables and graphs as often as from formulas.

The exam in one table

Described in our words. The College Board's exam page has the current question counts, timings and weights.

Section	Part	Calculator
I, multiple choice	Part A	Not allowed
I, multiple choice	Part B	Graphing calculator required
II, free response	Part A	Graphing calculator required
II, free response	Part B	Not allowed

The sections are weighted and combined into a score from 1 to 5. No AP question is reproduced on this page.

Where it meets MHF4U and precalculus (AP Precalculus vs precalculus)

The overlap table below reads the AP units against our MHF4U course map, sorted by its strands, and against the twelve precalculus ideas calculus assumes. Chapter 3 of the free playbook shows how to map a course like this before you study it.

AP unit	MHF4U	Precalculus	New in AP
1. Polynomial and rational	Strand C: polynomial functions, dividing and factoring, rational functions and asymptotes. Near-complete overlap	Polynomial behaviour, rational functions	Rates of change described in words
2. Exponential and logarithmic	Strand A: exponential functions, logarithms, the laws, equations, models	Exponentials and logarithms	Fitting models to data; sequences as discrete models
3. Trigonometric and polar	Strand B: radians, the unit circle, graphs, identities, equations	Trigonometry in radians, identities	Polar coordinates and polar graphs
4. Parameters, vectors, matrices	Not in MHF4U (vectors are in MCV4U)	Sometimes	Not examined

MHF4U's strand D, characteristics of functions (rates of change, combining functions, inverses, transformations), runs underneath all three AP units. MHF4U's map has the whole course in teaching order.

For a US reader: an ordinary precalculus course covers the same function families. AP Precalculus adds the exam, the modelling language and the polar functions. A school's own course may include topics the AP course does not examine, so do not assume your class notes and the AP framework match line for line.

The modelling habit: what the exam asks that a textbook does not

A textbook question usually gives you a formula and asks you to do something with it. AP Precalculus questions often give you a table or a short scenario instead, and ask three things: which function type fits, why, and what a rate of change means in the scenario's units.

The habit, in order:

1. Name the family from the pattern before any formula. Equal differences in the outputs over equal input steps: linear. Equal ratios: exponential. A repeating pattern: periodic, so trigonometric. Second differences equal: quadratic.
2. Say why in one sentence. "The values triple every step, so the function is exponential."
3. Interpret in the scenario's words. Not "the rate is 4" but "the tank's volume increases by about 4 litres per minute at t = 3".

The fault that costs most is a right formula with no interpretation. If the question gives units, the answer carries them.

Worked example: one problem per examined unit, with the common wrong line

These problems are ours, written for this page. They are not AP items.

Unit 1: a rational function

Find the holes and asymptotes of f(x) = (x² − 4)/(x² − x − 2).

Factor the numerator: x² − 4 = (x − 2)(x + 2)

Factor the denominator: x² − x − 2 = (x − 2)(x + 1)

Cancel the common factor (x − 2), for x ≠ 2: f(x) = (x + 2)/(x + 1)

Hole: at x = 2, the cancelled factor. Its height is (2 + 2)/(2 + 1) = 4/3, so the hole is at (2, 4/3).

Vertical asymptote: x = −1, where the remaining denominator is zero.

Horizontal asymptote: the degrees are equal and both leading coefficients are 1, so y = 1/1 = 1.

The common wrong line: calling x = 2 a vertical asymptote because it makes the original denominator zero. A factor that cancels makes a hole, not an asymptote. Asymptotes and rational functions works through more cases, and polynomial division handles the cases where the factors are not obvious.

Unit 2: a logarithmic equation

Solve log₂(x) + log₂(x − 2) = 3.

Combine with the product law: log₂(x(x − 2)) = 3

Rewrite in exponential form: x(x − 2) = 2³ = 8

Expand and rearrange: x² − 2x − 8 = 0

Factor: (x − 4)(x + 2) = 0, so x = 4 or x = −2

Check each in the original: log₂(4) + log₂(2) = 2 + 1 = 3, so x = 4 works. log₂(−2) is undefined, so x = −2 is rejected.

Answer: x = 4.

The common wrong line: keeping both solutions. The step that combined the logarithms allowed values the original equation does not, so every solution is checked against the original. Log rules and log equations explains why this happens.

The modelling twin: a table whose outputs triple at every step is exponential, and its rate of change grows in proportion to its value.

Unit 3: a trigonometric model

A Ferris wheel has a radius of 10 m, its centre is 12 m above the ground, and it makes one turn every 40 s. A rider starts at the bottom. Write the rider's height h, in metres, after t seconds.

Midline: the centre, 12 m.

Amplitude: the radius, 10 m.

Period: 40 s, so the coefficient of t is 2π/40 = π/20.

Start at the bottom: −cos starts at its minimum, so use h(t) = 12 − 10cos(πt/20).

Check t = 0: h(0) = 12 − 10cos(0) = 12 − 10 = 2 m, the bottom.

Check t = 20, half a turn: h(20) = 12 − 10cos(π) = 12 + 10 = 22 m, the top.

The common wrong line: h(t) = 12 + 10sin(πt/20) with no shift. Sine with no phase shift starts at the midline, so this rider would begin 12 m up, level with the centre, not at the bottom.

Checkable answers: the hole at (2, 4/3); x = −1; y = 1; x = 4; h(0) = 2 and h(20) = 22.

How to study for AP Precalculus: the method

• Procedures by worked example, then cold. Sweller and Cooper (1985) found that students who studied worked examples learned algebra procedures faster than students who solved the same problems unaided. Study one worked problem, close it, and do the next one alone.
• The function families as cards. Front: a pattern in a table. Back: the family, the reason, and a sample rate-of-change sentence. Domain, range, inverses and transformations is the reading to do beside these.
• Mixed sets from week 3. The exam does not say which family a question is about. Rohrer and Taylor (2007) found that practice shuffled across problem types tested better than practice blocked by type, even though it felt harder.
• One released free-response set a week, under time, self-scored against the published guidelines on AP Central.
• An error log by skill, not by chapter: factoring, asymptotes, log laws, trig in radians, interpretation sentences. The practice-test method and the error log is the system.
• Put the calculator down for the no-calculator parts. Do the algebra by hand in practice, or it will be slow on the day.

The six-week plan

Week	Recall task	Mixed set	Free response
1	Diagnostic: one released exam under time, the log by skill; unit 1 review	Unit 1 only	The diagnostic's, self-scored
2	Unit 2: log laws, equations, exponential models	Units 1–2	One set
3	Unit 3: trig in radians, models, polar graphs	Units 1–3	One set
4	The modelling habit on tables and scenarios across all three units	Units 1–3	One set
5	A second released exam; the top three skills from the log	The log's top three	The exam's, self-scored
6	The final week: light review, hardest skills early	One short set a day	None in the last two days

Roediger and Karpicke (2006) found that practising retrieval beat rereading on a test a week later, which is why each week carries a timed set rather than a reread of the notes.

Is AP Precalculus worth taking? (Is AP Precalculus hard?)

An honest answer. AP Precalculus is a precalculus course with an exam, so its difficulty is the ordinary course's plus the modelling language: naming families from data and writing interpretation sentences.

What it gives you is the functions calculus assumes, learned to an exam standard. That pays off directly in AP Calculus, the course after it.

What it is not is calculus. It is not a substitute for a calculus course, and the credit a college gives for it is the college's decision, not the College Board's.

A student who already has MHF4U covers most of it. For that student the exam is a check plus the polar unit, not a new course.

AP Precalculus, on one screen

• The four units: polynomial and rational; exponential and logarithmic; trigonometric and polar; parameters, vectors and matrices. The first three are examined.
• The exam: multiple choice and free response, each split into a part with a graphing calculator and a part without; scored 1 to 5.
• The overlap: MHF4U's strands C, A and B against units 1, 2 and 3; polar functions are the new part.
• The modelling habit: family from the pattern, the reason in a sentence, the answer in the scenario's words.
• The plan: worked examples then cold, mixed sets from week 3, a released set a week, a second full exam in week 5, a light final week.