VCE Foundation Mathematics cheat sheet

Categorical data names groups (nominal) or has a meaningful order (ordinal); numerical data is discrete (counted) or continuous (measured).

Mean = sum of values ÷ number of values; it uses every data point and is pulled toward outliers.

Median = middle value of ordered data (average the two middle values for even n); mode = most frequent value, possible in any data type.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1, resistant to outliers.

Outlier rule: a value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is flagged as an outlier.

For grouped data, estimate the mean using class midpoints: mean ≈ Σ(f × midpoint) ÷ Σf.

A frequency table lists values or classes with counts; relative frequency = frequency ÷ total, and relative frequencies sum to 1.

Two-way tables give joint, marginal and conditional counts; compare row percentages to describe association between variables.

Symmetric data: mean ≈ median. Skewed right: mean > median. Skewed left: mean < median.

Probability scale: 0 ≤ P(E) ≤ 1; P(not E) = 1 − P(E); sum of probabilities of all outcomes = 1.

P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events P(A and B) = 0.

For independent events P(A and B) = P(A) × P(B); for conditional probability P(A and B) = P(A) × P(B|A).

Equally likely outcomes: P(E) = number of favourable outcomes ÷ total number of outcomes.

Two-way table conditional probability: P(A|B) = count in A and B ÷ count in B (restrict to the row or column for B).

Expected frequency = probability × number of trials; it predicts long-run counts, not guaranteed results.

A simulation or experiment gives relative frequency ≈ probability; more trials generally means closer agreement.

Random sampling methods: simple random, systematic (every kth item), stratified (proportional to group size), cluster (whole groups).

Order of operations: brackets, indices, multiplication and division left to right, then addition and subtraction left to right.

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.

Expand using the distributive law a(b + c) = ab + ac; factorise by taking out the highest common factor.

Linear equation solving: apply the same operation to both sides until the variable is isolated; check by substitution.

Simultaneous equations: substitution or elimination; the solution is the point satisfying both equations.

Percentage change = (new − old) ÷ old × 100%; multiply by (1 + r) for an increase and (1 − r) for a decrease.

Simple interest I = Prt using decimal rate r; compound growth A = P(1 + r)^n.

Scientific notation a × 10^n with 1 ≤ a < 10; round using the required number of decimal places or significant figures.

A relation is a set of ordered pairs; a function assigns exactly one output to each input (passes the vertical line test).

Gradient between two points: m = (y2 − y1)/(x2 − x1); horizontal lines have m = 0, vertical lines have undefined gradient.

Linear form y = mx + c: m is the gradient, c is the y-intercept; parallel lines share m, perpendicular gradients multiply to −1.

Point–gradient form y − y1 = m(x − x1) builds an equation from one point and the gradient.

x-intercept: set y = 0 and solve; y-intercept: set x = 0. These are the axis crossing points.

Quadratic y = ax² + bx + c has a parabola graph; axis of symmetry x = −b/(2a) and turning point on that line.

Direct proportion y = kx graphs as a line through the origin; inverse proportion y = k/x graphs as a hyperbola.

Exponential growth/decay y = a·b^x: b > 1 grows, 0 < b < 1 decays; the y-intercept is a.

Transformations: y = f(x) + k shifts up k; y = f(x − h) shifts right h; y = a·f(x) stretches vertically by a.

Perimeter of a rectangle = 2(l + w); area of a rectangle = l × w; area of a triangle = ½ × base × height (height perpendicular to base).

Circle: circumference C = 2πr = πd; area A = πr²; arc length = (θ/360) × 2πr for angle θ in degrees.

Volume: prism/cylinder V = base area × height; cone V = ⅓πr²h; sphere V = (4/3)πr³.

Surface area of a rectangular prism = 2(lw + lh + wh); cylinder curved area = 2πrh, total = 2πrh + 2πr².

Pythagoras: in a right triangle a² + b² = c², where c is the hypotenuse; use it for diagonals and distances.

Units: 1 m = 100 cm, 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³, 1 L = 1000 mL = 1000 cm³.

Similar figures: matching angles equal and side lengths in a fixed ratio k; areas scale by k² and volumes by k³.

Angle facts: angles on a straight line sum to 180°, angles at a point sum to 360°, angles in a triangle sum to 180°.

Density = mass ÷ volume; speed = distance ÷ time; rearrange to find the unknown quantity.

Five-number summary: minimum, Q1, median, Q3, maximum; the box plot displays these with a box from Q1 to Q3.

Standard deviation measures spread about the mean; a larger value means data is more spread out.

When comparing groups, report both a centre (mean or median) and a spread (range, IQR or standard deviation).

Back-to-back stem plots or parallel box plots compare two distributions on the same scale.

Time series data is plotted against time; look for trend (long-term direction) and seasonality (regular repeating pattern).

Misleading graphs: truncated axes, uneven intervals or 3D effects exaggerate differences; check scale and starting value.

Class intervals must be equal width and non-overlapping; midpoint = (lower + upper) ÷ 2.

Correlation describes the strength and direction of a linear association; it does not prove causation.

Two-step experiments: list outcomes with a tree diagram or sample space grid, then count favourable outcomes.

Without replacement, the denominator and favourable counts decrease after each draw; with replacement they stay the same.

Venn diagrams: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); the complement region is outside the sets.

Mutually exclusive events cannot occur together; independent events have P(A|B) = P(A).

Test independence by checking whether P(A and B) equals P(A) × P(B) for the observed data.

Sample space for two dice has 36 equally likely outcomes; for one die, 6 outcomes each with probability 1/6.

Relative frequency from a table estimates probability; the law of large numbers says estimates improve with more trials.

Bias: a sample drawn only from a convenient group may not represent the population; random selection reduces bias.

Linear inequalities: solve like equations, but reverse the inequality sign when multiplying or dividing by a negative number.

Substitution: replace each variable with its value and evaluate using order of operations.

Rearranging formulas: perform inverse operations to isolate the required variable, treating other letters as constants.

Expanding (a + b)(c + d) = ac + ad + bc + bd; recognise perfect squares and difference of two squares.

Solving quadratics: factorise, or use the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a.

Ratio problems: split a total by adding ratio parts, then multiply the total by each part over the sum of parts.

Rounding and estimation: round each value to 1 significant figure to check whether an answer is reasonable.

Direct and inverse variation: y = kx means y/x is constant; y = k/x means xy is constant.

To find where two graphs meet, set the expressions equal and solve for x, then substitute to find y.

Gradient from a graph = rise ÷ run, using two clear points on the line.

A line perpendicular to one with gradient m has gradient −1/m.

Distance between two points: d = √((x2 − x1)² + (y2 − y1)²); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

Sketching: label intercepts, turning points and any asymptotes before drawing the curve.

Step graphs and piecewise rules apply different formulas over different intervals of the domain.

Interpreting graphs: gradient represents a rate (speed, cost per unit); area under a line may represent a total.

Domain and range describe allowable inputs and resulting outputs; state them for each model in context.

Composite shapes: split into rectangles, triangles and circles, then add or subtract areas.

Sector area = (θ/360) × πr²; use the same fraction for arc length of the same circle.

Trigonometry in right triangles: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Find an unknown side with trig ratios and an unknown angle with inverse trig (sin⁻¹, cos⁻¹, tan⁻¹).

Bearings are measured clockwise from north and written with three digits, e.g. 045°.

Scale drawings: actual length = drawing length × scale factor; convert units before calculating.

Total surface area sums every exposed face; for closed solids include top and bottom.

Nets fold into solids; count faces, edges and vertices to check the shape of a net.