Functional Relationships

Direct proportional

Formula: y = mx (or y ∝ x) Example: A = abc in Beer's law — absorbance is directly proportional to concentration. As x increases, y increases by the same factor.

Inverse proportional

Formula: y = m/x (or y ∝ 1/x) Example: PV = nRT — at fixed n and T, as pressure rises, volume falls. As x increases, y decreases by the same factor.

Quadratic / parabola (power)

Formula: y = mx² Example: rate = k[A]² — double the concentration and the rate quadruples. As x doubles, y quadruples; as x triples, y becomes nine times larger.

Inverse square

Formula: y = m/x² Example: v = R(1/n₁² − 1/n₂²) — as n rises, the term drops roughly as an inverse square. As x doubles, y drops to one-fourth; as x triples, y drops to one-ninth.

Logarithmic

Formula: y = ln(x) Example data points: (1, 0), (2, 0.7), (3, 1.1) Rises quickly at first, then flattens out.

Exponential

Formula: y = 2ˣ (or y = bˣ) Example data points: (1, 2), (2, 4), (3, 8), (4, 16) Slow at first, then climbs steeply — doubling with every step.

Reading data points

Direct: (1,2) (2,4) (3,6) Inverse: (1,1) (2,½) (3,⅓) Inverse square: (1,1) (2,¼) (3,⅑) Parabola: (1,1) (2,4) (3,9) Logarithmic: (1,0) (2,0.7) (3,1.1) Exponential: (1,2) (2,4) (3,8) (4,16)

Exam habit

Formula: recognize the relationship type from the equation or graph Example: rate = k[A]² tells you the rate depends on concentration squared. You are always given equations, constants, and conversions — what you must supply is recognizing the relationship.