Grade 11 math

Add the terms of an arithmetic sequence with Sₙ = n(a₁ + aₙ)/2, write sums compactly in sigma notation, and use the formula for counting and money problems.

How to build a model by adding, subtracting or dividing functions you know, why adding functions adds their graphs, and how composition feeds one function into another.

Why mathematicians introduced i with i² = −1, how to add, subtract and multiply complex numbers, how the powers of i cycle, and how the conjugate makes division possible.

Interest compounded n times a year, what happens as n grows without bound, the number e ≈ 2.718, and continuous growth with A = Pe^(rt).

Model quantities that vary together: direct variation y = kx, inverse variation y = k/x, joint and combined variation, and finding the constant of variation.

How polynomial long division mirrors division of numbers, how to write a quotient as q(x) + r(x)/b(x), and why synthetic division is a shortcut for dividing by x − a.

Test whether a function is even, odd or neither by computing f(−x), and see the answer in its graph: a mirror image in the y-axis or a half turn about the origin.

Model decay, growth and cooling with exponential functions, solve for the time with logarithms, and compare values on pH, decibel and earthquake scales.

Factor polynomials beyond quadratics: sums and differences of cubes, factoring by grouping, and polynomials in quadratic form, then use the factors to solve equations.

A polynomial of degree n has exactly n complex zeros, counted with multiplicity. Use that count and conjugate pairs to find every zero and build polynomials.

How to derive the sum of a finite geometric series by multiplying by the ratio and subtracting, and how the same sum sets a savings total and a monthly loan payment.

How to sketch a polynomial from its factored form: where it crosses the x-axis, why a repeated factor makes it bounce, and how the leading term decides both ends.

How factoring a rational function reveals its zeros, holes and vertical asymptotes, why the graph shoots off near an asymptote, and how degrees decide the end behavior.

How unrolling the unit circle produces a wave, what amplitude, period, midline and phase shift each control, and how to fit a sine model to a repeating situation.

Graph y = tan x with its period π and vertical asymptotes, and the reciprocal functions secant, cosecant and cotangent built from sine and cosine.

Graph y = bˣ and y = log_b x: growth and decay, asymptotes, domain and range, the mirror image in y = x, and shifts and stretches of both.

Add all the terms of a geometric sequence: when −1 < r < 1 the sum is a/(1 − r). Use it to turn repeating decimals into fractions and total a bouncing ball's path.

Find the best value of a linear objective over a region cut out by linear inequalities, and see why the maximum and minimum always sit at a corner.

What a logarithm is, why it is the inverse of an exponential, how to use logs to solve for an unknown exponent, and how to evaluate any logarithm with a calculator.

Solve polynomial and rational inequalities with a sign chart: find every critical number, test each interval, and handle zeros of the denominator correctly.

What makes an equation an identity, how to prove one, how identities generate Pythagorean triples and mental-math shortcuts, and how Pascal's triangle expands (x + y)ⁿ.

The three logarithm rules for products, quotients and powers, why they follow from the exponent rules, expanding and condensing logs, and change of base.

Solve inequalities like x² − x − 6 > 0 by finding where the parabola crosses the x-axis, testing the intervals between, and writing the answer as intervals.

How the quadratic formula gives complex solutions when the discriminant is negative, why they come in conjugate pairs, and what the Fundamental Theorem of Algebra says.

Why every solving step is a claim that follows from the one before, why squaring or clearing denominators can add false answers, and how to solve and check both kinds.

How rational expressions follow the rules of fractions, why only common factors cancel, how to multiply and divide them, and how to add them over a common denominator.

List the possible rational zeros of a polynomial, test them with synthetic division, and use each zero you find to factor and solve the polynomial completely.

Why dividing p(x) by x − a leaves the remainder p(a), why p(a) = 0 means x − a is a factor, and how the two theorems turn one known root into a full factorization.

Simplify square roots and cube roots, multiply and divide radicals, add like radicals, and rationalize denominators, including with conjugates.

Solve equations with the unknown in an exponent or inside a logarithm: matching bases, taking logs of both sides, condensing logs, and rejecting extraneous solutions.

Solve equations like 2 sin x = 1: find one angle, use the unit circle to find every solution in one cycle, and add multiples of the period to get them all.

The shapes of y = √x and y = ∛x, why one starts at a point and the other runs both ways, and how shifts and stretches move the starting point, center, domain and range.

Solve linear systems in x, y and z by elimination: reduce to two equations in two unknowns, solve, substitute back, and see why a solution is where three planes meet.

Why the Pythagorean theorem on the unit circle gives sin²θ + cos²θ = 1, what makes it an identity, and how to find sine, cosine or tangent from one value and a quadrant.

Why the right-triangle definition stops at 90°, how the unit circle replaces it, the exact values at π/6, π/4 and π/3, and why sine and cosine repeat.

How to add vectors end to end and by components, why the length of a sum is usually less than the sum of the lengths, how subtraction works, and what scaling does.

Tell a circle, ellipse, parabola and hyperbola apart from Ax² + Cy² + Dx + Ey + F = 0, then complete the square to find the center, radius, vertices and axes.

Raise a complex number to a power by raising its length and multiplying its angle, then reverse the rule to find all n of its nth roots, evenly spaced on a circle.

Derive sin 2θ, cos 2θ and tan 2θ from the sum formulas, reverse them into half-angle and power-reducing forms, and use them for exact values and equations.

Why an ellipse is every point whose distances to two foci add to a constant, how squaring twice turns that into x²/a² + y²/b² = 1, and how the hyperbola follows.

Why sine, cosine and tangent need a restricted domain before they can be undone, what arcsin, arccos and arctan return, and how to solve trig equations in context.

Prove a statement for every positive integer with two steps: check it for n = 1, then show that whenever it holds for k it holds for k + 1.

How matrices store data, how to add, scale and multiply them, why the order of multiplication matters, and how the determinant decides whether an inverse exists.

How a 2×2 matrix moves every vector in the plane, why its columns show where the whole plane goes, and why the determinant measures how it scales area.

Describe a curve by giving x and y as functions of a parameter t: plotting and direction, eliminating the parameter, line segments and projectile motion.

Locate points by distance and angle, convert between polar and rectangular coordinates, and graph polar equations: circles, cardioids and roses.

How to write a system of linear equations as a single matrix equation AX = B, why multiplying by the inverse solves it, and what a zero determinant says about the system.

Why cos(α − β) = cos α cos β + sin α sin β, proved by rotating a chord of the unit circle, how the other formulas follow, and how to find exact values like cos 15°.

How to plot complex numbers, why rectangular and polar form name the same point, why multiplying multiplies lengths and adds angles, and how to find distances.

Multiply two vectors into a number: u · v = u₁v₁ + u₂v₂ equals |u||v|cos θ, which measures the angle between them, tests for right angles and gives projections.

What makes a quantity a vector, how to draw and name one, why its components come from subtracting the start from the end, and how to use vectors for velocity.

Prove that two trigonometric expressions agree for every angle: work one side at a time, rewrite in sines and cosines, and trade with the Pythagorean identities.

Find the probability of exactly k successes in n independent trials with the binomial formula, see why it works, and answer at least and at most questions.

How reshuffling an experiment's results shows what chance alone can produce, how to judge whether a gap between two groups is significant, and how to read a data report.

How to find the expected payoff of a game, compare strategies such as insurance plans, make choices fair with randomness, and read test results with false positives.

Why two honest samples give different answers, how simulating a thousand samples reveals the spread, where the square-root rule comes from, and how to read a poll.

The counting principle, factorials, when order matters and when it does not, why a combination divides a permutation by r!, and how to use counts to find probabilities.

How a random variable turns outcomes into numbers, how to build its probability distribution from theory or from data, and why expected value is the long-run average.

What each kind of study can and cannot show, where randomization enters, why random selection and random assignment do different jobs, and how confounding works.

Why the area of any triangle is one half ab sin C, derived by dropping a height from a vertex, why the formula still works when the angle is obtuse, and how to use it.

Why c² = a² + b² − 2ab cos C is the Pythagorean theorem with a correction, proved with coordinates, and how to find a side from SAS or an angle from SSS.

Why each side of a triangle over the sine of its opposite angle gives the same number, how to solve triangles with it, and why SSA can give two triangles or none.

Why a parabola is every point equally far from a point and a line, how to derive its equation from that definition with the distance formula, and what the focus does.