A logarithm is the inverse of exponentiation. logₐ(b)=c means a raised to the power c equals b.

Upward-opening parabola vertex form: y = a(x-h)² + k, where a > 0 and (h,k) is the vertex.

Fundamental Theorem of Algebra: a polynomial of degree n has exactly n roots (counting multiplicity), which may be real or complex.

A hole occurs when a factor cancels from both numerator and denominator. The function is undefined at that x-value but has no asymptote.

Euler's identity: e^(iπ) + 1 = 0. Often called the most beautiful equation in mathematics, connecting e, i, π, 1, and 0.

Polar coordinates (r,θ): r is the distance from the origin (pole), θ is the angle from the positive x-axis.

De Moivre's Theorem: [r(cosθ+i sinθ)]ⁿ = rⁿ(cos(nθ)+i sin(nθ)). Used to raise complex numbers to powers.

Asymptotes of horizontal hyperbola: y = ±(b/a)x. The hyperbola approaches but never crosses these lines.

Descartes' Rule of Signs: the number of positive real zeros equals the number of sign changes in f(x), or less by an even number.

Conjugate of a+bi is a-bi. Multiplying a complex number by its conjugate gives a real number: a²+b².

Pascal's Triangle gives the binomial coefficients for expanding (a+b)ⁿ. Row n gives coefficients for (a+b)ⁿ.

For an ellipse: c² = a² - b², where c is the distance from center to each focus.

Rectangular to polar: r = √(x²+y²), θ = arctan(y/x). Adjust θ for the correct quadrant.

Geometric sequence: aₙ = a₁ × r^(n-1). Multiply first term by ratio raised to (n-1).

Phase shift of y = sin(bx-c) = c/b to the right. Factor out b: sin(b(x - c/b)).

Ellipse: x²/a² + y²/b² = 1. The semi-major axis is the larger of a and b.

Transpose A^T swaps rows and columns: the element in row i, column j moves to row j, column i.

A matrix is invertible (non-singular) if and only if its determinant ≠ 0.

Determinant of 2×2 matrix: det = ad - bc. Multiply diagonals and subtract.

A polynomial of degree n: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀, where aₙ ≠ 0.