Algebra & Functions Beginner

A linear equation has the form ax + b = 0, solved by isolating x = -b/a. Graphically it is a straight line y = mx + c with slope m and intercept c. Slope is rise over run: (y₂−y₁)/(x₂−x₁).

A quadratic ax² + bx + c = 0 is solved by the formula x = (−b ± √(b²−4ac)) / 2a. The discriminant b²−4ac tells you the number of real roots: positive → two, zero → one, negative → none (complex).

Key families: linear, polynomial, exponential a·bˣ (growth/decay), and logarithmic (the inverse of exponential). Exponentials model compound interest and population; logs compress wide ranges (decibels, pH, Richter scale).

Indices: xᵃ·xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1. Logs invert: log(xy) = log x + log y, log(xⁿ) = n·log x. These rules turn multiplication into addition — the basis of slide rules and many engineering shortcuts.