ORELA Mathematics (NT304) cheat sheet

Mathematics

• Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)] / (2a); the discriminant b² − 4ac gives two real roots if positive, one if zero, and a conjugate pair of complex roots if negative.

• Slope of a line through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have slopes whose product is −1 (negative reciprocal).

• Point-slope form y − y₁ = m(x − x₁) and slope-intercept y = mx + b are equivalent; standard form Ax + By = C has slope −A/B.

• Distance between two points is √[(x₂ − x₁)² + (y₂ − y₁)²]; the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

• Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, and a^(m/n) = ⁿ√(aᵐ).

• Logarithm laws: log(xy) = log x + log y, log(x/y) = log x − log y, log(xᵖ) = p log x, and log_b b = 1, log_b 1 = 0; change of base log_b x = ln x / ln b.

• Function composition (f ∘ g)(x) = f(g(x)); an inverse f⁻¹ exists only if f is one-to-one, and then f(f⁻¹(x)) = x with graphs mirrored across y = x.

• Arithmetic sequence: aₙ = a₁ + (n − 1)d, sum Sₙ = n(a₁ + aₙ)/2; geometric sequence: aₙ = a₁rⁿ⁻¹, sum Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1, and infinite sum a₁/(1 − r) when |r| < 1.