VCE General Mathematics cheat sheet

Index laws: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1 (a ≠ 0); a^(−n) = 1/a^n.

Logarithm laws: log(ab) = log a + log b; log(a/b) = log a − log b; log(a^n) = n log a; log_b(b^x) = x.

Expansion: (a + b)^2 = a^2 + 2ab + b^2; (a − b)^2 = a^2 − 2ab + b^2; (a + b)(a − b) = a^2 − b^2.

Quadratic formula: x = [−b ± √(b^2 − 4ac)]/(2a) solves ax^2 + bx + c = 0; the discriminant Δ = b^2 − 4ac gives two, one, or no real roots as Δ > 0, Δ = 0, Δ < 0.

Simultaneous linear equations: solve by substitution or elimination; the system has no solution when the lines are parallel (same gradient, different intercepts).

Direct proportion y = kx and inverse proportion y = k/x, where k is found from a known data pair.

Exponential growth/decay: y = A·b^t with b > 1 for growth and 0 < b < 1 for decay; the rule of 70 estimates doubling time as 70 ÷ percentage growth rate.

Percentage change = (new − old)/old × 100%; successive changes multiply, so a 20% rise then a 20% fall gives 1.2 × 0.8 = 0.96 of the original.

A relation is a function if every x-value maps to exactly one y-value; the vertical line test detects this on a graph.

Linear function y = mx + c: m is the gradient (rise ÷ run) and c is the y-intercept; parallel lines share m, perpendicular lines have m1·m2 = −1.

Gradient between two points: m = (y2 − y1)/(x2 − x1).

Quadratic y = ax^2 + bx + c is a parabola; the axis of symmetry is x = −b/(2a), and the turning point lies on that line.

Turning-point (vertex) form y = a(x − h)^2 + k has vertex (h, k); a > 0 opens upward, a < 0 opens downward.

Transformations: y = f(x) + k shifts up by k; y = f(x − h) shifts right by h; y = a·f(x) dilates vertically by factor a; y = f(−x) reflects in the y-axis.

Inverse functions undo the original; find by swapping x and y and solving, and the graph of f^(−1) is a reflection of f in the line y = x.

Exponential y = a·b^x has a horizontal asymptote at y = 0; logarithmic y = log_b(x) is defined only for x > 0 and is the inverse of the exponential.

Domain restrictions come from denominators (≠ 0) and even roots (radicand ≥ 0).

Perimeter and area: rectangle A = lw; triangle A = ½bh; parallelogram A = bh; trapezium A = ½(a + b)h; circle C = 2πr, A = πr^2.

Volume: prism/cylinder V = area of base × height; pyramid/cone V = ⅓ × base area × height; sphere V = (4/3)πr^3.

Surface area of a cylinder = 2πr^2 + 2πrh; surface area of a sphere = 4πr^2; surface area of a cone = πr^2 + πrl (l = slant height).

Pythagoras: c^2 = a^2 + b^2 in a right triangle; the converse tests for a right angle.

Trigonometry in right triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; use inverse functions to find angles.

Sine rule: a/sin A = b/sin B = c/sin C; cosine rule: c^2 = a^2 + b^2 − 2ab·cos C; area of any triangle = ½ab·sin C.

Similar figures: corresponding lengths scale by k, areas by k^2, and volumes by k^3.

Unit conversion: 1 cm^2 = 100 mm^2, 1 m^2 = 10 000 cm^2, 1 m^3 = 1 000 000 cm^3, and 1 mL = 1 cm^3.

Density = mass ÷ volume; speed = distance ÷ time, rearranged as distance = speed × time.