Precalculus

126 core topics + 65 prerequisite topics taught as needed · approximately 48 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

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 · 4 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

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.

Radicals & Exponentials · 5 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

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.

Functions & Algebra II · 13 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Domain & Range	Which inputs a function accepts, and which outputs it can produce.
Function Composition	Feeding one function's output into another: f(g(x)).
Inverse Functions	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Transformations of Functions	How f(x − h) + k slides a graph around the plane.
Systems by Elimination	Adding or subtracting equations to cancel a variable.
Polynomial Division	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials	Far from the origin, only the leading term matters.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

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