AP Precalculus

160 core topics + 50 prerequisite topics taught as needed · approximately 53 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph — topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. A cumulative quiz follows every 6 lessons. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Radicals & Exponentials · 10 topics

Product Rule for Exponents [E]	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules [E]	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents [M]	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Scientific Notation [M]	Writing very large or very small numbers as c × 10ⁿ.
Simplifying Radicals [M]	Pulling perfect-square factors out of a square root.
Operations with Radicals [M]	Adding like radicals and multiplying square roots.
Rational Exponents [M]	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Radical Equations [H]	Isolate the radical, then square both sides.
Exponential Growth & Decay [M]	Quantities that multiply by the same factor each time step: y = a·bᵗ.
Compound Interest [H]	Money growing exponentially: A = P(1 + r)ᵗ.

Functions & Algebra II · 24 topics

Function Notation & Evaluation [E]	Reading f(x) notation and plugging in inputs.
Domain & Range [E]	Which inputs a function accepts, and which outputs it can produce.
Function Composition [M]	Feeding one function's output into another: f(g(x)).
Inverse Functions [M]	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Transformations of Functions [M]	How f(x − h) + k slides a graph around the plane.
Piecewise Functions [M]	Functions defined by different rules on different intervals.
Absolute Value Equations [M]	|x − a| = b splits into two linear equations.
Systems by Elimination [H]	Adding or subtracting equations to cancel a variable.
Nonlinear Systems [H]	Where a line meets a parabola: set the two formulas equal.
Complex Numbers [M]	The imaginary unit i = √(−1) and numbers of the form a + bi.
Operations with Complex Numbers [M]	Multiplying complex numbers with FOIL and i² = −1.
Quadratics with Complex Roots [H]	When the discriminant is negative, the roots come in a conjugate pair a ± bi.
Polynomial Division [H]	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems [M]	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials [H]	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials [E]	Far from the origin, only the leading term matters.
Simplifying Rational Expressions [M]	Factor top and bottom, then cancel the common factor.
Operations on Rational Expressions [H]	Multiplying and dividing algebraic fractions.
Rational Equations [H]	Clearing denominators to solve equations with x below the line.
Logarithms [M]	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms [M]	Logs turn products into sums, quotients into differences, powers into multiples.
Exponential & Log Equations [M]	Matching bases and rewriting between exponential and log form.
Arithmetic Sequences [M]	Sequences that grow by a constant difference each step.
Geometric Sequences [M]	Sequences that grow by a constant ratio each step.

Trigonometry · 15 topics

Right-Triangle Trigonometry [M]	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Solving for Sides with Trig [M]	Using a known angle and one side to find another side.
Degrees & Radians [E]	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle [M]	Exact sine, cosine, and tangent values at the special angles.
Trig of Any Angle [H]	Reference angles plus quadrant signs extend trig beyond 90°.
Graphs of Sine & Cosine [M]	Reading amplitude, period, and midline from y = a sin(bx) + c.
Phase Shifts & Other Trig Graphs [M]	Horizontal (phase) shifts of trig graphs, and the period of tangent.
The Pythagorean Identity [M]	sin²θ + cos²θ = 1 links sine and cosine of the same angle.
Basic Trig Identities [M]	Quotient, reciprocal, and even-odd identities.
Sum & Difference Formulas [H]	Expanding sin(A ± B) and cos(A ± B) to reach non-special angles.
Double-Angle Formulas [H]	sin 2x = 2 sin x cos x and cos 2x = 1 − 2 sin²x.
Trig Equations [H]	Isolating a trig function and reading solutions off the unit circle.
Law of Sines [H]	In any triangle, each side over the sine of its opposite angle is constant.
Law of Cosines [H]	c² = a² + b² − 2ab cos C generalizes the Pythagorean theorem.
Inverse Trig Functions [M]	arcsin, arccos, and arctan undo the trig functions on restricted ranges.

Precalculus · 12 topics

Asymptotes of Rational Functions [M]	Where rational functions blow up and where they level off.
Graphs of Rational Functions [M]	Holes, asymptotes, and intercepts tell the whole story of the graph.
Polynomial Inequalities [M]	Sign charts: zeros split the number line into test intervals.
Vectors: Components & Magnitude [M]	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations [M]	Scaling, adding, and dotting vectors — all component by component.
Parametric Equations [M]	Describing a moving point by giving x and y as functions of time.
Polar Coordinates [H]	Locating points by distance from the origin and angle from the x-axis.
Polar Graphs [M]	Recognizing circles, lines, and rose curves from polar equations.
Circles & Ellipses [M]	Reading centers, radii, and intercepts from conic equations.
The Binomial Theorem [H]	Expanding (x + a)ⁿ without multiplying it out term by term.
Sigma Notation & Series [M]	Σ compresses a sum: read the limits, add up the terms.
Average Rate of Change [M]	The slope of the secant line: (f(b) − f(a)) / (b − a).

Precalculus: Vectors & Matrices · 13 topics

Vector Components from Two Points [E]	Head minus tail, coordinate by coordinate.
Magnitude of a Vector [E]	A vector's length comes straight from the Pythagorean theorem.
Adding & Subtracting Vectors [E]	Vectors combine component by component — tip-to-tail in coordinates.
Scalar Multiples & Combinations [E]	A scalar stretches every component; combinations mix scaled vectors.
The Dot Product [M]	Multiply matching components and add — two vectors in, one number out.
Perpendicular Vectors [M]	Two vectors are perpendicular exactly when their dot product is zero.
Classifying the Angle Between Vectors [M]	The sign of the dot product tells acute, right, or obtuse.
2×2 Matrices: Addition & Scalar Multiples [E]	Same-shape matrices add entry by entry; a scalar hits every entry.
Multiplying a Matrix by a Vector [M]	Each output component is a row of the matrix dotted with the vector.
2×2 Matrix Multiplication [M]	Row of the left matrix times column of the right, entry by entry.
The 2×2 Determinant [M]	Down-diagonal product minus up-diagonal product: ad − bc.
Determinant as Parallelogram Area [M]	The parallelogram on ⟨a, b⟩ and ⟨c, d⟩ has area |ad − bc|.
2×2 Systems as Matrix Equations [H]	A pair of linear equations is one matrix equation with one solution.

Precalculus: Conic Sections · 12 topics

Circle Equations: Center & Radius [E]	Read the center and radius straight off (x − h)² + (y − k)² = r².
Circle Through a Given Point [M]	The radius is the distance from the center to any point on the circle.
Circles by Completing the Square [M]	Turn x² + y² + Dx + Ey + F = 0 back into center–radius form.
Parabolas: Focus & Directrix [M]	In x² = 4py the focus sits p above the vertex and the directrix p below.
Parabolas with a Shifted Vertex [M]	Vertex, focus, and directrix stay p apart no matter where the vertex sits.
Ellipses in Standard Form [E]	The larger denominator points along the major axis: a² under it, b² under the other.
Foci of an Ellipse: c² = a² − b² [M]	The foci sit inside the ellipse on the major axis, c² = a² − b² from center.
Hyperbolas in Standard Form [E]	The positive squared term tells you the axis the two branches open along.
Foci of a Hyperbola: c² = a² + b² [M]	Hyperbola foci sit beyond the vertices: c² adds a² and b².
Asymptotes of a Hyperbola [M]	The branches hug the lines y = ±(b/a)x through the center.
Eccentricity as an Exact Fraction [M]	e = c/a measures shape: below 1 for ellipses, above 1 for hyperbolas.
Classifying a Conic from Its Equation [M]	Compare the squared terms: their signs and coefficients name the conic.

Precalculus: Polar Coordinates & Complex Numbers · 10 topics

Adding & Subtracting Complex Numbers [E]	Combine real parts and imaginary parts separately.
Multiplying Complex Numbers [M]	FOIL, then replace i² with −1.
Powers of i [E]	The powers of i repeat every four steps.
Modulus of a Complex Number [M]	The distance from the origin: √(a² + b²).
Complex Conjugates [M]	Flip the sign of i; the product is real.
Dividing Complex Numbers [M]	Multiply top and bottom by the denominator's conjugate.
Polar to Rectangular [M]	x = r cos θ, y = r sin θ.
Exact Radical Coordinates [M]	The 30° and 60° coordinates carry a √3.
Rectangular to Polar (r) [M]	The radius is the distance to the origin.
Reference Angles [M]	The acute angle to the nearest x-axis.

Precalculus: Trig Identities & Equations in Action · 12 topics

Recovering tan θ from One Ratio [M]	Combine the Pythagorean identity with tan θ = sin θ / cos θ, then let the quadrant fix the sign.
Secant & Cosecant from One Ratio [M]	Find the missing ratio by the Pythagorean identity, then flip it: sec = 1/cos, csc = 1/sin.
Cotangent from One Ratio [M]	cot θ = cos θ / sin θ — the quotient identity read the other way up.
Exact Values by Decomposition [H]	Split an unusual angle into a sum of special angles, then expand.
Combining Two Known Angles [H]	Given sines of two angles, build sin(A ± B) and cos(A ± B) as exact fractions.
Computing sin(2x) [M]	sin 2x = 2 sin x cos x — recover the missing factor with its correct sign first.
Computing cos(2x) [M]	cos 2x = 1 − 2 sin²x = 2 cos²x − 1 — one squared ratio is enough.
Identities That Collapse to a Number [M]	sec²−tan² = 1, csc²−cot² = 1, and each function times its reciprocal is 1.
Simplifying to One Function [M]	Rewrite a product or quotient in terms of sine and cosine, then cancel.
Counting Solutions on [0°, 360°) [M]	Each attainable value of sine or cosine is hit twice per turn — except at the peaks.
The Smallest Solution in Degrees [M]	Isolate the function, find the reference angle, then take the least angle in range.
Verifying the Right Formula [M]	Spot the correct expansion and reject the near-miss sign and swap errors.

Precalculus: Exponential & Logarithmic Functions · 13 topics

Evaluating Exponential Functions [E]	Plug integer inputs into f(x) = a·bˣ — including 0 and negatives.
Graphs: y-Intercept & Asymptote [M]	y-intercept a, growth when b > 1, decay when 0 < b < 1, floor at y = 0.
Evaluating Logarithms [M]	log_b(x) asks: to what power must b be raised to get x?
Log Laws in Computation [M]	Logs turn products into sums, quotients into differences, powers into multiples.
Exponentials & Logs as Inverses [M]	log_b and b^x undo each other — reflections across the line y = x.
Exponential Equations: Same Base [M]	Match the bases, then set the exponents equal.
Exponential Equations with Logs [M]	Take a logarithm of both sides to bring the exponent down.
The Natural Base e and ln [M]	e ≈ 2.718 is the natural base; ln is log base e, its exact inverse.
Exponential Growth & Decay Models [M]	Model y = a·bᵗ: multiply the start by the factor once per time step.
Continuous Compound Interest [M]	Compounding at every instant uses the natural base: A = P·e^(rt).
Doubling Time [M]	Doubling every T means y = a·2^(t/T) — count the doublings first.
Half-Life [M]	Half-life is the time to halve once — y = a·(1/2)^(t/H).
Solving Logarithmic Equations [M]	Rewrite log_b(expr) = k as expr = b^k, then solve.

Precalculus: Rational Functions, Series & the Binomial Theorem · 13 topics

Vertical Asymptotes vs. Holes [M]	A cancelling factor makes a hole; a surviving denominator factor makes an asymptote.
Horizontal Asymptotes by Degree [M]	Compare the top and bottom degrees to read off the horizontal asymptote.
Intercepts of Rational Functions [M]	x-intercepts come from the numerator's zeros; the y-intercept is f(0).
The Coordinates of a Hole [M]	Cancel the common factor, then plug the x-value into what remains.
Combinations C(n, r) [M]	Count unordered selections with C(n, r) = n! / (r!(n − r)!).
Permutations P(n, r) [M]	Count ordered arrangements with P(n, r) = n! / (n − r)!.
Permutation or Combination? [M]	Decide whether order matters, then pick P(n, r) or C(n, r).
Pascal's Triangle Entries [M]	Every Pascal entry is a C(n, k), and each row sums to 2ⁿ.
Extracting a Binomial Coefficient [H]	One term of (x ± a)ⁿ: C(n, k)·aⁿ⁻ᵏ, with the sign tracked.
Finite Arithmetic Series [M]	Sum an arithmetic series with S = n(first + last)/2.
Finite Geometric Series [M]	Sum n geometric terms with S = a(1 − rⁿ)/(1 − r).
Evaluating Sigma Notation [M]	Read the limits, apply the standard sum formulas, and add.
Infinite Geometric Series [M]	When |r| < 1 the endless sum converges to a/(1 − r).

Precalculus: Trigonometric Graphs & Models · 13 topics

Amplitude of a Sinusoid [E]	Amplitude is |a|, half the vertical distance from crest to trough.
Period in Radians: 2π / b [M]	In radians the period of a sinusoid is 2π divided by b.
Period in Degrees: 360 / b [E]	In degrees the period of a sinusoid is 360 divided by b.
The Midline y = d [E]	The midline is y = d, the horizontal center the wave swings around.
Vertical Shift of a Sinusoid [E]	Adding d shifts the whole graph up (d > 0) or down (d < 0) by |d|.
Phase Shift of a Sinusoid [M]	Factor the b out: sin(bx − c) shifts right by c/b, not by c.
Period of y = tan(bx) [M]	Tangent repeats twice as fast as sine: its period is π / b.
Maximum & Minimum Values [M]	Max is midline + amplitude; min is midline − amplitude.
Amplitude from Max & Min [M]	Amplitude is half the gap between the highest and lowest values.
Midline from Max & Min [M]	The midline sits at the average of the highest and lowest values.
Modeling: Finding the Amplitude [M]	Turn a periodic phenomenon's high and low into an amplitude.
Modeling: Finding the Midline [M]	The model's midline is the center height or the average of high and low.
Modeling: Finding the Period [M]	The period is the time for one full cycle of the phenomenon.

Precalculus: Parametric & Polar Applications · 13 topics

Evaluating a Parametric Path [E]	Plug a value of t into each equation to locate the moving point.
Eliminating the Parameter: Lines [M]	Solve x = t + b for t, substitute, and read off slope and intercept.
Eliminating the Parameter: Parabolas [M]	A squared parameter eliminates into a quadratic in x — expand carefully.
Eliminating the Parameter: Circles [M]	x = a cos t, y = a sin t squares and adds to x² + y² = a².
Displacement Along a Parametric Path [M]	Displacement in a coordinate is its ending value minus its starting value.
Distance Between Two Positions [M]	The distance between two positions is √(Δx² + Δy²).
Polar to Rectangular: Exact Radicals [M]	At 30°, 45°, and 60° one coordinate carries an exact radical.
Polar to Rectangular: Whole Coordinates [M]	Half the special-angle conversions land on a plain rational coordinate.
Rectangular to Polar: the Radius [M]	The polar radius is the distance from the origin, √(x² + y²).
The Modulus of a Point [M]	A point's modulus is its polar radius: √(x² + y²).
The Circle r = a [E]	When r is a constant, every angle gives the same distance — a circle.
The Circle r = a cos θ [M]	r = a cos θ is an off-center circle of radius a/2 through the origin.
Classifying Polar Graphs [M]	Sort r = a, θ = c, r = a cos θ, and r = a cos(nθ) by their shapes.

Prerequisite material — taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 6 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)	Same-denominator addition.
Adding Fractions (Unlike Denominators)	Rewrite over a common denominator first.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction	Line up the decimal points.
Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Multi-Step Equations	Equations needing distribution or variables on both sides.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 6 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Systems of Equations (Substitution)	Two equations, two unknowns.
Arithmetic Sequences	Add the same amount each step.
Geometric Sequences	Multiply by the same ratio each step.

Quadratics & Polynomials · 10 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides — remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
Vertex of a Parabola	The turning point at x = −b/2a.

Geometry · 8 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Distance & Midpoint	Measuring segments in the coordinate plane.
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

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