AP Precalculus

329 core topics + 56 prerequisite topics taught as needed · approximately 114 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. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. 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]	The vertical asymptotes and the end behavior (horizontal or slant asymptote) of a rational function.
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 canceling 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.

Precalculus - Deep II · 37 topics

Trig Equations: a Solution in Radians [H]	Read the reference angle, place it in the right quadrants, and report the asked solution.
Trig Equations: Sum of the Solutions [H]	Add the two solutions on the interval - symmetry often makes the sum a tidy multiple of pi.
Quadratic Trig Equations: Counting Roots [H]	Solve the quadratic in sine or cosine, then count how many angles each root gives.
Half-Angle Exact Values [H]	cos(theta/2) = sqrt((1 + cos theta)/2); the sign follows the half-angle's quadrant.
Combining a Sine and Cosine: R sin(x + φ) [H]	a sin x + b cos x is one sinusoid with amplitude R = sqrt(a^2 + b^2).
The Tangent Sum & Difference Formula [H]	tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
The Law of Sines [H]	Sides are proportional to the sines of their opposite angles: a/sin A = b/sin B.
Law of Cosines: Finding a Side [H]	c^2 = a^2 + b^2 - 2ab cos C uses two sides and the included angle.
Law of Cosines: Finding an Angle [H]	Rearrange to cos C = (a^2 + b^2 - c^2)/(2ab) to recover an angle from three sides.
The Ambiguous (SSA) Case [H]	Compare the opposite side a to the height b sin A: it decides 0, 1, or 2 triangles.
Area from Two Sides and the Included Angle [H]	Area = (1/2) a b sin C - half the product of two sides times the sine of the angle between.
Polar Equations to Rectangular [H]	Multiply through by r and use r^2 = x^2 + y^2, r cos = x, r sin = y.
Rectangular Equations to Polar [H]	Substitute x = r cos, y = r sin; circles become r = a, and lines become r cos = k or theta = c.
Recognizing Polar Curves [H]	r = a(1 + cos) is a cardioid; r = a + b cos is a limacon; the nth-multiple angle marks a rose.
Work as a Dot Product [H]	Work is the dot product of force and displacement: W = Fx dx + Fy dy.
The Angle Between Two Vectors [H]	cos of the angle is the dot product over the product of the magnitudes.
Perpendicular & Parallel Vectors [H]	Perpendicular means the dot product is 0; parallel means one is a scalar multiple of the other.
Direction Angle of a Vector [H]	The direction angle points the vector; equal-magnitude components land on the 45-degree diagonals.
Parametric Ellipses [H]	x = a cos t, y = b sin t squares and adds to x^2/a^2 + y^2/b^2 = 1.
Speed Along a Parametric Line [H]	For linear motion the speed is sqrt((dx/dt)^2 + (dy/dt)^2), constant in time.
Projectile Height from Parametric Motion [H]	The height y = v0 t - 5 t^2 is a downward parabola; its vertex gives the peak.
Area of an Ellipse [H]	An ellipse encloses pi a b, the two semi-axes multiplied (a generalized pi r^2).
Matching a Conic to Its Equation [H]	Equal same-sign squares are a circle; unequal an ellipse; opposite signs a hyperbola; one square a parabola.
Polynomial End Behavior [H]	The leading term's degree parity and sign fix both tails of the graph.
Zeros & Their Multiplicities [H]	A factor's exponent is the zero's multiplicity; the total is the degree, and parity decides cross vs touch.
Slant (Oblique) Asymptotes [H]	When the numerator's degree is one more than the denominator's, long division gives a slant line.
Intercepts of a Rational Function [H]	x-intercepts are the numerator's zeros; the y-intercept is f(0).
Sum of an Arithmetic Series [H]	S_n = (n/2)(2 a_1 + (n - 1) d), or n times the average of the first and last terms.
Sum of a Finite Geometric Series [H]	S_n = a_1 (r^n - 1)/(r - 1) for a common ratio r not equal to 1.
Sum of an Infinite Geometric Series [H]	When |r| < 1 the series converges to a_1 / (1 - r).
The nth Term of a Sequence [H]	Arithmetic: a_1 + (n - 1) d. Geometric: a_1 r^(n - 1).
The Constant Term of a Binomial Expansion [H]	In (a x + c/x)^n the general term is C(n,k)(a x)^(n-k)(c/x)^k; the power of x is zero when k = n/2.
A Specific Term of a Binomial Expansion [H]	The coefficient of x^j in (x + a)^n is C(n, j) a^(n - j) - no need to expand fully.
Limits at a Removable Discontinuity [H]	Cancel the common factor, then substitute - the hole's height is the limit.
Limits at Infinity of Rational Functions [H]	Equal top and bottom degrees give the leading-coefficient ratio; a smaller top gives 0.
Estimating a Limit Numerically [H]	If both sides of a table close in on the same value, that value is the limit.
Limits from a Graph [H]	The limit watches the approach from both sides, not the plotted point's value.

Precalculus - Function Analysis & Transformations (Deep) · 43 topics

Operations on Functions at a Point [H]	Evaluate each function at the input first, then combine those two outputs with the indicated operation.
A Coefficient of the Product fg [H]	Collect every pair of terms whose exponents add to the requested power, then add those products.
Excluded Values of a Quotient f/g [H]	The quotient f/g is undefined wherever g is zero and wherever either function is already undefined.
The Domain of a Sum [H]	The domain of f + g is the overlap of the two domains: satisfy the radical condition and avoid the value that makes the denominator zero.
Evaluating a Composition [H]	Evaluate the inner function first, then apply the outer function to that output.
Solving a Composition Equation [H]	Undo the outer function first to learn what the inner output must be, then solve the inner equation.
Domain of a Rational Composition [H]	Exclude every input where the inner function is undefined and every input whose output is barred from the outer function's domain.
Domain of a Radical Composition [H]	Require the outer radicand to be non-negative, or strictly positive when the radical sits in a denominator, then solve that inequality for x.
Composition from a Table of Values [H]	Read the inner output from its column, then look that number up in the x column to read the outer output.
Decomposing h into f of g [H]	The inner function is the expression the outer operation is applied to; the outer function is that operation acting on a single placeholder.
Recovering the Outer Function [H]	Set u equal to the inner formula, solve for x in terms of u, and substitute to express h entirely in u; that expression is f.
A Value of the Inverse [H]	Finding f inverse of k means solving f(x) = k for x, since the inverse reverses the roles of input and output.
Verifying an Inverse Formula [H]	A candidate is the inverse exactly when composing it with f in either order returns x.
The Inverse of a Rational Function [H]	Set the rational expression equal to the target value, clear the denominator, and collect the x-terms on one side.
Inverses on a Restricted Domain [H]	Solving the quadratic equation gives two candidate inputs; the restricted domain selects which one is the inverse value.
Domain and Range of an Inverse [H]	An inverse swaps the two sets: the range of f inverse is the domain of f, and the domain of f inverse is the range of f.
Even, Odd, or Neither [H]	Replace x by negative x and compare: matching f gives an even function, matching the opposite of f gives an odd function, neither gives neither.
Using Symmetry to Find Values [H]	Symmetry transfers a known value across zero: an even function repeats it and an odd function reverses its sign.
Parity of Products and Compositions [H]	Track the sign that each factor or layer contributes when x is replaced by negative x, then multiply those signs together.
The Image of a Point [H]	Vertical constants act directly on the y-coordinate, while horizontal constants act on the x-coordinate in the opposite sense.
A Sequence of Transformations [H]	Translate each described step into the constant it contributes, keeping horizontal shifts inside the parentheses and vertical changes outside.
Building and Evaluating a Transformed Function [H]	Write the described sequence as an equation in the template A times the parent of B times x minus C, plus D, then substitute the input.
Extreme Values on a Closed Interval [H]	Compare the values at the endpoints with the value at the vertex, using the vertex only when it lies inside the interval.
How a Transformation Moves a Zero [H]	A zero survives every vertical stretch or reflection; only the horizontal replacement of x changes where it occurs.
Evaluating a Piecewise Function [H]	Decide which condition the input satisfies, then use only that branch's rule.
Making a Piecewise Rule Continuous [H]	Continuity at the boundary requires the two branch formulas to produce the same output there, which is one equation in the unknown constant.
The Corner of an Absolute Value Graph [H]	The corner occurs where the expression inside the absolute value equals zero, since that is where the branch definition switches.
Greatest-Integer Arithmetic [H]	The greatest integer function rounds toward negative infinity, so a negative non-integer moves down to the next integer.
A Step-Function Pricing Model [H]	Charges that jump at fixed intervals are modeled with a ceiling: any part of an interval counts as a whole one.
Average Rate of Change of a Rational Function [H]	Evaluate the function at both endpoints, subtract the outputs, and divide by the change in the input.
Comparing Average Rates from a Table [H]	Divide each change in output by its own change in input, then compare the resulting slopes rather than the raw changes.
Average Rate of Change in Context [H]	The secant slope carries the units of output per unit of input, so the computation answers a question about an average speed, rate, or yield.
Recovering a Missing Endpoint [H]	Write the average rate of change with the unknown endpoint in it, set that expression equal to the given rate, and solve.
Simplifying the Difference Quotient [H]	Expand f(x + h), subtract f(x) so every surviving term contains h, then divide each term by h.
The Difference Quotient of a Rational Function [H]	Combine the two rational outputs over a common denominator first; the numerator then contains a factor of h that cancels.
The End-Behavior Model of a Rational Function [H]	For large inputs only the leading terms matter, so the function behaves like the quotient of those two leading terms.
Where a Graph Crosses Its Horizontal Asymptote [H]	A horizontal asymptote describes end behavior only, so setting the function equal to that value and solving locates any crossing.
One-Sided Behavior at a Vertical Asymptote [H]	Test the sign of each factor at an input just to the chosen side of the asymptote; the combined sign decides which way the graph goes.
The Value Missing from the Range [H]	Solving y = f(x) for x fails at exactly one output, the height of the horizontal asymptote, so that value is missing from the range.
Area as a Function of One Variable [H]	Use the fencing or perimeter constraint to write the second dimension in terms of the first, then multiply the two dimensions.
The Open-Box Volume Model [H]	Cutting a square of side x from each corner shortens both base dimensions by twice x and makes x the height.
Composing a Geometric Model with Time [H]	Substitute the time-dependent radius into the geometric formula to obtain the quantity as a single function of time.
Revenue and Profit Models [H]	Revenue is price times quantity demanded, and profit subtracts both the per-item cost and the fixed cost from that revenue.

Precalculus - Trigonometric Functions, Identities & Equations (Deep) · 45 topics

Coterminal Angles in Radians [H]	Add or subtract whole multiples of 2 pi until the angle lands in the required window.
Degree and Radian Conversion Beyond the Special Angles [H]	Multiply degrees by pi/180 to get radians, and radians by 180/pi to get degrees.
Arc Length from a Degree Measure [H]	Arc length is s = r theta with theta in radians, so convert the degree measure first.
The Central Angle from an Arc [H]	Divide the arc length by the radius to get the central angle in radians.
Angular Speed in Radians per Second [H]	Angular speed is angle swept per unit time; one revolution is 2 pi radians.
Linear Speed on a Rotating Wheel [H]	Linear speed is v = r omega: the rim travels the radius times the angle swept.
Two Pulleys on One Belt [H]	A belt forces equal linear speeds, so r1 omega1 = r2 omega2 and the smaller pulley spins faster.
Coordinates of a Terminal Point [H]	The terminal point of theta on the unit circle is (cos theta, sin theta), signed by quadrant.
Recovering the Angle from a Terminal Point [H]	Read the reference angle from the size of the coordinates, then let their signs pick the quadrant.
Exact Secant, Cosecant, and Cotangent Values [H]	Take the reciprocal of the matching sine, cosine, or tangent value, then rationalize.
Locating the Quadrant from Two Signs [H]	Two sign conditions pin down one quadrant: all positive in I, sine in II, tangent in III, cosine in IV.
Combining Several Exact Values [H]	Evaluate each special value with its quadrant sign, then combine like radical terms.
Choosing b from a Required Period [H]	Period = 2 pi / b, so b = 2 pi divided by the required period.
Choosing c from a Required Phase Shift [H]	In sin(bx - c) the shift is c/b, so a shift of h needs c = b h.
Evaluating a Sinusoid Exactly [H]	Substitute, simplify the argument to a special angle, then scale by the amplitude and add the shift.
The End of One Full Cycle [H]	A cycle runs from argument 0 to argument 2 pi, so it ends at x = (c + 2 pi)/b.
Counting Crossings, Peaks, and Intercepts [H]	On 0 to 2 pi a sinusoid runs b full cycles, so features repeat b times.
A Ferris Wheel Model: Height at a Given Time [H]	Evaluate the height model by turning the elapsed time into a fraction of one revolution.
When a Model First Reaches an Extreme [H]	Maxima and minima alternate every half period, so step half a period from a known extreme.
Solving a Sinusoidal Model for a Time [H]	Write the depth as M + A cos(2 pi t / P), set it equal to the target, and solve for t.
Writing the Equation of a Sinusoidal Model [H]	Amplitude is half the range, the midline is the average of the extremes, and b is 2 pi over the period.
Asymptotes of a Tangent Graph [H]	Tangent is undefined, and its graph has a vertical asymptote, where its argument equals pi/2 plus a multiple of pi.
Counting Asymptotes of Secant and Cosecant [H]	Secant breaks where cosine is zero and cosecant breaks where sine is zero.
x-Intercepts of Tangent and Cotangent Graphs [H]	Tangent is zero where sine is zero; cotangent is zero where cosine is zero.
The Range Gap of a Cosecant or Secant Graph [H]	Since |csc| and |sec| are at least 1, the range is everything except the open interval of width 2|a| about d.
Principal Values of arcsin(sin x) and Friends [H]	An inverse function returns the angle in its own principal range, which need not be the angle you started with.
Compositions with an Inverse Trig Function [H]	Let the inverse name an angle, sketch its reference triangle, then read off the requested ratio.
Double Angles of an Inverse Trig Value [H]	Name the inverse value as an angle, find its sine and cosine, then apply the double-angle formula.
Domains and Ranges of the Inverse Trig Functions [H]	arcsin and arccos accept inputs in [-1, 1]; their outputs live in [-pi/2, pi/2] and [0, pi].
Complementary Sums of Inverse Values [H]	arcsin k + arccos k = pi/2 for every k in [-1, 1], and arctan k + arctan(1/k) = pi/2 for k > 0.
Secant from Tangent and Back [H]	Dividing sin^2 + cos^2 = 1 by cos^2 gives 1 + tan^2 = sec^2; by sin^2 it gives 1 + cot^2 = csc^2.
Simplifying a Trigonometric Fraction [H]	Replace a Pythagorean block with its single-function equivalent, then cancel.
Cofunction Identities [H]	Each function of an angle equals its cofunction of the complement: sin theta = cos(90 degrees - theta).
Identity or Conditional Equation? [H]	An identity holds for every defined value; a conditional equation holds only at particular angles.
Reading a Sum Formula Backwards [H]	Recognize the expanded pattern and compress it into one function of the combined angle.
Exact Tangent Values by Decomposition [H]	Split the angle into two special angles, apply tan(A + B), then rationalize the denominator.
The Tangent Double-Angle Formula [H]	tan 2 theta = 2 tan theta / (1 - tan^2 theta).
Half-Angle Values with Nested Radicals [H]	sin(theta/2) = plus or minus sqrt((1 - cos theta)/2), with the sign set by the half angle's quadrant.
Power-Reduction Formulas [H]	sin^2 theta = (1 - cos 2 theta)/2 and cos^2 theta = (1 + cos 2 theta)/2.
The Key Step in Verifying an Identity [H]	Verify by transforming one side alone: expand, write everything in sines and cosines, then combine.
Multiple-Angle Equations: a Requested Solution [H]	Solve for the whole argument bx over its own longer interval, then divide every solution by b.
Counting Solutions of a Multiple-Angle Equation [H]	Compressing the graph by a factor of b multiplies the number of solutions on a fixed interval by b.
Equations Needing a Double-Angle Substitution [H]	Rewrite the doubled angle in terms of the single angle, then factor the resulting equation.
The General Solution of a Trig Equation [H]	Add 2 k pi to each solution on one revolution, except for tangent, which repeats every pi.
Solving a Shifted Trig Equation [H]	Solve for the shifted argument first, then add the shift back and keep only the solutions in the interval.

Precalculus - Vectors, Polar Form, Parametrics & Matrices (Deep) · 44 topics

The Unit Vector in a Given Direction [H]	Dividing a vector by its own magnitude produces the unit vector pointing the same way.
Component Form from Magnitude and Direction [H]	A vector of magnitude m at direction angle theta has components m cos theta and m sin theta.
The Magnitude of a Scalar Multiple [H]	Scaling a vector by k multiplies its length by the absolute value of k.
The Magnitude of a Resultant Force [H]	Add the forces component by component, then take the length of the sum.
The Direction Angle of a Resultant [H]	Add the forces first, then read the direction angle of the sum from its components.
The Equilibrant of a Force System [H]	The equilibrant is the opposite of the resultant, so it has the same magnitude and reversed components.
Components from a Compass Bearing [H]	A bearing is measured clockwise from north, so the east component uses sine and the north component uses cosine.
Ground Speed and Wind in Navigation [H]	Ground velocity is air velocity plus wind velocity, and ground speed is the magnitude of that sum.
The Exact Angle Between Two Vectors [H]	Compute the cosine from the dot product and the magnitudes, then name the angle it belongs to.
The Magnitude of a Sum from the Dot Product [H]	Expanding |u + v|^2 as |u|^2 + 2(u dot v) + |v|^2 links lengths to the dot product.
The Scalar Projection onto a Vector [H]	The scalar projection of u onto v is the dot product divided by the length of v.
The Vector Projection and the Orthogonal Part [H]	The vector projection scales v by (u dot v) over the square of the length of v.
Weight Components on a Ramp [H]	On a ramp inclined theta, weight splits into w sin theta along the ramp and w cos theta into it.
Writing a Vector as a Combination of Two Others [H]	Matching components in a u + b v = w produces a two-by-two system for the coefficients.
Rectangular to Polar: the Angle [H]	The polar angle comes from the reference angle of |y/x| placed in the quadrant of the point.
Equivalent Polar Representations of a Point [H]	Adding 180 degrees to the angle reverses the sign of the radius, and adding 360 degrees changes nothing.
Where a Polar Curve Passes Through the Pole [H]	Set r equal to zero and solve the resulting trigonometric equation for theta.
Classifying a Limacon by the Ratio a/b [H]	For r = a plus or minus b times a cosine, the ratio a/b decides inner loop, cardioid, dimpled, or convex.
Counting the Petals of a Rose Curve [H]	A rose r = a cos(n theta) has n petals when n is odd and 2n petals when n is even.
Greatest and Least Distance from the Pole [H]	Since a sine or cosine ranges from -1 to 1, r = a plus or minus b trig ranges from a - b to a + b.
The Distance Between Two Polar Points [H]	Two radii and the angle between them form a triangle, so the law of cosines gives the distance.
The Argument of a Complex Number [H]	The argument is the direction angle of a + bi, found from the reference angle and the quadrant.
Trigonometric Form of a Complex Number [H]	Trigonometric form writes z as r(cos theta + i sin theta) with r the modulus and theta the argument.
Products in Trigonometric Form [H]	Multiplying complex numbers in trigonometric form multiplies the moduli and adds the arguments.
Quotients in Trigonometric Form [H]	Dividing complex numbers in trigonometric form divides the moduli and subtracts the arguments.
De Moivre's Theorem: Powers in Trigonometric Form [H]	Raising r(cos theta + i sin theta) to the nth power raises the modulus to the nth power and multiplies the argument by n.
De Moivre's Theorem in Rectangular Form [H]	Convert to trigonometric form, apply De Moivre, then convert the power back to a + bi.
The nth Roots of a Complex Number [H]	The n roots share the modulus r to the power 1/n and their arguments start at theta/n and step by 360/n degrees.
Eliminating the Parameter: Hyperbolas [H]	Secant and tangent parametrizations collapse to a hyperbola through sec squared minus tan squared equals one.
The Range of a Coordinate on a Parametric Path [H]	Because sine and cosine stay between -1 and 1, a coordinate a cos t + h stays between h - a and h + a.
Initial Velocity Components of a Projectile [H]	A launch speed v at angle theta splits into a horizontal v cos theta and a vertical v sin theta.
The Time of Flight of a Projectile [H]	Setting the height equation equal to zero and solving for t gives the moment the projectile lands.
The Horizontal Range of a Projectile [H]	The range is the horizontal velocity multiplied by the total time of flight.
The Height at a Given Horizontal Distance [H]	Solve the horizontal equation for the time at that distance, then substitute into the height equation.
Where a Parametric Path Crosses an Axis [H]	A path meets the y-axis where x(t) = 0 and the x-axis where y(t) = 0, so solve that equation for t first.
The Inverse of a 2x2 Matrix [H]	The inverse of [[a, b], [c, d]] swaps a and d, negates b and c, and divides everything by the determinant.
Singular Matrices: When No Inverse Exists [H]	A square matrix has an inverse exactly when its determinant is nonzero.
The 3x3 Determinant [H]	Expand along a row using cofactors: each entry multiplies the 2x2 determinant left after deleting its row and column.
Cramer's Rule for a 2x2 System [H]	Each variable equals the determinant of the coefficient matrix with that column replaced by the constants, divided by the determinant.
Cramer's Rule in a 3x3 System [H]	One variable of a three-variable system is a ratio of two 3x3 determinants.
Solving a Matrix Equation with an Inverse [H]	Multiplying both sides of A X = B by the inverse of A on the left isolates X.
The Dimensions of a Matrix Product [H]	A product AB exists only when the columns of A match the rows of B, and it has the rows of A and the columns of B.
Multiplying Matrices Beyond 2x2 [H]	The entry in row i and column j of a product is the dot product of row i of the left matrix with column j of the right.
Back-Substitution from Row-Echelon Form [H]	Row-echelon form solves the last variable immediately, and each earlier row then yields one more.

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.

Limits & Continuity · 4 topics

Limits: Graphical & Numerical	What value a function approaches - which need not be the value it takes.
One-Sided Limits	Approaching from the left or right - and when the two disagree.
Infinite Limits & Vertical Asymptotes	A limit of +∞ or −∞ at a vertical asymptote; the sign of the shrinking denominator decides which.
Limits at Infinity	The value a function approaches as x goes to +∞ or −∞; for a rational function, compare the degrees of the top and bottom.

Infinite Series · 2 topics

Convergence of Sequences (BC)	A sequence converges if aₙ approaches a limit.
Geometric Series (BC)	Σarⁿ = a/(1−r) when |r| < 1.

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