QCE Maths Methods flashcards

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

Discriminant

For the quadratic ax²+bx+c=0, the discriminant is b²−4ac. It is positive for two distinct real roots, zero for one repeated root, and negative for no real roots.

Remainder theorem

When a polynomial P(x) is divided by (x−a), the remainder equals P(a). Consequently, (x−a) is a factor exactly when P(a)=0.

Change of base rule

For positive bases not equal to one, logₐx equals log_b x divided by log_b a. This allows logarithms in any base to be evaluated using common or natural logs.

Completing the square

Rewriting ax²+bx+c as a(x+b/2a)² + c − b²/4a. This form reveals the turning point and axis of symmetry of the parabola and helps solve quadratics.

Absolute value inequality

For b>0, the inequality |x−a|<b means a−b<x<a+b, while |x−a|>b means x<a−b or x>a+b. The expression inside is a distance from a.

One-to-one function

A function where each output value corresponds to exactly one input value. Only one-to-one functions have inverse functions, and they pass both vertical and horizontal line tests.

Inverse function graph

The graph of f⁻¹ is the reflection of the graph of f across the line y=x. The domain of f becomes the range of f⁻¹, and the range becomes the domain.

Composite function domain

For f(g(x)) to be defined, x must lie in the domain of g and g(x) must lie in the domain of f. The composite is generally not equal to g(f(x)).

Odd and even functions

An even function satisfies f(−x)=f(x) and is symmetric about the y-axis. An odd function satisfies f(−x)=−f(x) and has rotational symmetry about the origin.

Horizontal asymptote of rational function

Compare degrees of numerator and denominator. If the denominator degree is larger, the asymptote is y=0; if degrees are equal, it is the ratio of leading coefficients.

Vertical dilation and translation

For y=af(x)+k, the graph is stretched vertically by factor a and then translated k units up. Dilations are applied before translations when reading transformations from the equation.

Derivative first principles

The derivative is the limit as h approaches zero of [f(x+h)−f(x)] divided by h. It represents the instantaneous rate of change or gradient of the tangent.

Chain rule

The derivative of f(g(x)) equals f′(g(x)) multiplied by g′(x). It is used for composite functions such as sin(3x), e^{x²} and ln(2x+1).

Quotient rule

For u/v, the derivative is (u′v−uv′)/v². The numerator order matters, and the rule applies only where v is not zero.

Stationary point classification

At a stationary point f′(x)=0. If f″(x)>0 the point is a local minimum; if f″(x)<0 it is a local maximum; if f″(x)=0 further testing is needed.

Definite integral as area

The definite integral from a to b of f(x) gives signed area between the curve and the x-axis. Regions below the axis contribute negatively, so geometric area requires absolute values.

Antiderivative of 1/x

The integral of 1/x with respect to x is ln|x| plus a constant. The absolute value is needed because the expression is defined for negative x as well.

Binomial distribution conditions

A binomial model requires a fixed number of independent trials, each with the same probability of success, and each trial having exactly two outcomes. Mean is np and variance np(1−p).

Expected value of discrete variable

The expected value E(X) equals the sum of each value multiplied by its probability. It represents the long-run average of the random variable over many repetitions.

Variance formula

Variance equals E(X²) minus the square of E(X). The standard deviation is the square root of the variance and has the same units as the variable.

Standardising a normal value

The z-score equals (x−μ)/σ. It states how many standard deviations a value lies from the mean and allows probabilities to be read from the standard normal distribution.

Correlation coefficient

A number between −1 and 1 describing the strength and direction of linear association. Values near ±1 indicate strong linear association; zero indicates no linear association.

Coefficient of determination

The value r², the square of the correlation coefficient, is the proportion of variation in the response variable explained by the least-squares linear regression on the explanatory variable.

Radian and degree conversion

Multiply degrees by π/180 to obtain radians, and multiply radians by 180/π to obtain degrees. A full revolution is 2π radians, equal to 360 degrees.

Arc length formula

For a circle of radius r and angle θ measured in radians, arc length equals rθ. Sector area equals half r squared times θ.

Pythagorean identity

For any angle θ, sin²θ+cos²θ=1. Dividing by cos²θ gives tan²θ+1=sec²θ, and dividing by sin²θ gives 1+cot²θ=cosec²θ.

Period of sine and cosine

The functions sin(bx) and cos(bx) have period 2π/b. The tangent function tan(bx) has period π/b, because tangent repeats every half revolution.

Amplitude of trigonometric graph

For y=a sin(bx)+d, the amplitude is |a|, the distance from the central axis y=d to a maximum or minimum. Amplitude is always non-negative.

Natural logarithm definition

The natural logarithm ln x is the logarithm to base e, where e is approximately 2.71828. It is the inverse of the exponential function eˣ.

Exponential function asymptote

The graph of y=aˣ has horizontal asymptote y=0 and y-intercept 1. Adding k gives asymptote y=k, and the range becomes (k,∞) when a is positive.

Half-life formula

For decay modelled by y=Ae^{−kt} with k>0, the half-life is ln2 divided by k. After each half-life the quantity falls to half its previous value.

Logarithmic function domain

The function logₐx is defined only for x>0, has range all real numbers, and has a vertical asymptote at x=0. Its graph passes through the point (1,0).