VCE General Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Mean

The arithmetic average: add all data values and divide by the number of values. It is affected by outliers because every value contributes to the total.

Median

The middle value when data are ordered. With an even number of values, average the two central values. It is resistant to outliers.

Interquartile range

The difference between the third and first quartiles, IQR = Q3 − Q1. It measures the spread of the middle 50% of the data and ignores extreme values.

Standard deviation

A measure of typical spread about the mean, roughly the average distance of values from the mean. Larger values indicate more variable data.

z-score

The number of standard deviations a value lies from the mean, computed as z = (x − x̄)/s. Positive means above the mean, negative below.

Least-squares regression line

The line minimising the sum of squared vertical residuals, written y = a + bx. It always passes through the point (x̄, ȳ).

Residual

The vertical difference between an actual data value and the value predicted by the regression line. A curved residual pattern signals a poor linear fit.

Conditional probability

The probability of event A given that B has occurred, written P(A|B) and equal to P(A and B) divided by P(B), provided P(B) > 0.

Independent events

Two events are independent when the occurrence of one does not change the probability of the other, so P(A and B) = P(A) × P(B).

Complement rule

The probability that an event does not occur equals 1 minus the probability that it does: P(not A) = 1 − P(A).

Index laws

Rules for combining powers: multiply by adding indices, divide by subtracting them, raise a power to a power by multiplying, and a^(−n) = 1/a^n.

Quadratic formula

For ax^2 + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b^2 − 4ac)]/(2a). It solves any quadratic equation.

Discriminant

The expression b^2 − 4ac for a quadratic. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.

Difference of two squares

The factorisation a^2 − b^2 = (a + b)(a − b). It converts a subtraction of squares into a product of two binomial factors.

Logarithm definition

log_b(x) is the exponent to which base b must be raised to obtain x; equivalently log_b(x) = y means b^y = x.

Direct proportion

A relationship of the form y = kx where k is a constant. Doubling x doubles y, and the graph is a straight line through the origin.

Inverse proportion

A relationship of the form y = k/x where k is constant. As x increases y decreases, and the graph is a hyperbola.

Simultaneous equations

Two or more equations solved together to find values satisfying all at once. Substitution or elimination is used; parallel lines give no solution.

Exponential growth and decay

Models of the form y = A·b^t. When b > 1 the quantity grows; when 0 < b < 1 it decays by a constant percentage each period.

Percentage change

The change expressed relative to the original value: (new − old)/old × 100%. Successive percentage changes multiply rather than add.

Function

A relation in which each input value maps to exactly one output value. A graph passes the vertical line test only if it is a function.

Gradient

The slope of a line, calculated as rise over run or (y2 − y1)/(x2 − x1) between two points. It measures the rate of change.

Perpendicular lines

Two lines meeting at right angles. Their gradients multiply to −1, so each gradient is the negative reciprocal of the other.

Axis of symmetry

The vertical line x = −b/(2a) that divides a parabola into mirror halves. The turning point of the parabola lies on this line.

Turning point form

Writing a quadratic as y = a(x − h)^2 + k, which displays the vertex at (h, k). The sign of a determines whether it opens up or down.

Inverse function

A function that reverses another, found by swapping x and y and solving. Its graph is the reflection of the original in y = x.

Asymptote

A line that a graph approaches but never reaches. Exponential graphs of the form y = a·b^x have the horizontal asymptote y = 0.

Domain restriction

Values excluded from a function's input set, arising where a denominator equals zero or where an even root's radicand is negative.

Pythagoras' theorem

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides: c^2 = a^2 + b^2.

Sine rule

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its matching angle. Used when given a matching side and angle pair.

Cosine rule

c^2 = a^2 + b^2 − 2ab·cos C relates three sides and one angle. Use it with two sides and the included angle, or with all three sides.

Similar figures scale factors

When lengths scale by factor k, areas scale by k^2 and volumes by k^3. This links the size change to surface area and capacity changes.