NCEA L1 Maths cheat sheet

Quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) solves ax² + bx + c = 0; the discriminant b² - 4ac gives the number of real roots (positive: two, zero: one, negative: none).

Sine rule: a/sin A = b/sin B = c/sin C; use when given two angles and any side, or two sides and a non-included angle.

Cosine rule: a² = b² + c² - 2bc cos A; use when given two sides and the included angle, or all three sides.

Area of a triangle = ½ab sin C, where C is the included angle between sides a and b.

For a straight line, gradient m = (y₂ - y₁)/(x₂ - x₁); equation forms y = mx + c and y - y₁ = m(x - x₁).

Exponential growth/decay: A = A₀(1 ± r)^t, where r is the rate per period as a decimal and t is the number of periods.

Probability of independent events: P(A and B) = P(A) × P(B); P(A or B) = P(A) + P(B) - P(A and B).

Mean = sum of values ÷ number of values; median = middle value of ordered data; mode = most frequent value.

Standard deviation measures spread: for a sample, s = √[Σ(x - x̄)²/(n - 1)]; for a population, σ = √[Σ(x - μ)²/N].

A census measures every member of a population; a sample measures a subset and is used when a census is impractical.

A simple random sample gives every individual an equal chance of selection; a stratified sample selects randomly within subgroups to ensure representation.

Bias occurs when a sampling method systematically favours some outcomes; a larger sample reduces variability but does not remove bias.

For a normal distribution, approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

The z-score z = (x - μ)/σ gives how many standard deviations a value lies from the mean.

A box plot displays the median, lower quartile (Q1), upper quartile (Q3), and extremes; the interquartile range IQR = Q3 - Q1 measures the middle 50% spread.

An outlier is often defined as a value below Q1 - 1.5×IQR or above Q3 + 1.5×IQR.

Correlation describes the strength and direction of a linear relationship; correlation does not imply causation.

A line of best fit (least-squares regression) predicts the response variable from the explanatory variable; the slope indicates the predicted change per unit increase.