MSCE Additional Mathematics key terms

The value b² − 4ac taken from a quadratic ax² + bx + c. Its sign tells you whether the equation has two real roots, one repeated real root, or no real roots.

Rewriting a quadratic in the form a(x + h)² + k by adding and subtracting a constant. It reveals the turning point of the graph and can solve equations without factorising.

A polynomial f(x) has (x − a) as a factor exactly when f(a) = 0. It links factors of a polynomial to its roots and supports synthetic or long division.

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). This gives a quick way to find remainders without performing full division.

Splitting a single rational expression into a sum of simpler fractions with linear or quadratic denominators. Useful for integration and for expanding expressions into series.

A statement comparing two expressions using <, ≤, > or ≥. Solving it gives a range of values, often written with interval notation or on a number line.

The function |x| that returns the non-negative size of x, so |x| = x when x ≥ 0 and |x| = −x when x < 0. Its graph is V-shaped with a vertex at the origin.

A function formed by applying one function to the output of another, written fg(x) or f(g(x)). It exists only when the range of the inner function fits the domain of the outer one.

The function f⁻¹ that reverses f, so f⁻¹(f(x)) = x for every x in the domain. Its graph is the reflection of y = f(x) in the line y = x.

The domain is the set of input values a function accepts; the range is the set of output values it actually produces. Both must be stated when defining a function fully.

A function where each output value comes from exactly one input, so no two inputs share an output. Only such functions have a true inverse that is also a function.

Changing a graph by translation, stretch or reflection. Knowing how y = f(x) becomes y = f(x − a) + b, y = af(x) or y = f(−x) lets you sketch quickly.

The turning point of a parabola, where the curve changes direction. For y = a(x − h)² + k the vertex is (h, k), giving the maximum or minimum value.

The steepness of a line, measured as the change in y divided by the change in x between two points. Parallel lines share a gradient; perpendicular gradients multiply to −1.

For a circle with centre (a, b) and radius r, the equation is (x − a)² + (y − b)² = r². Expanding it gives the general form x² + y² + 2gx + 2fy + c = 0.

The line that cuts a segment in half at right angles. Every point on it is equidistant from the segment's endpoints, which makes it useful for finding circle centres.

A sequence where each term differs from the previous one by a fixed constant d. The nth term is a + (n − 1)d and the sum of n terms is n/2(2a + (n − 1)d).

A sequence where each term is a fixed multiple r of the previous one. The nth term is arⁿ⁻¹, and the sum to infinity exists only when |r| < 1.

The constant amount added to each term to get the next in an arithmetic progression. It can be negative, giving a decreasing sequence.

The limiting total of an infinite geometric series, equal to a/(1 − r) when |r| < 1. If |r| ≥ 1 the series diverges and no finite sum exists.

The behaviour of a sequence or series whose terms approach a fixed limiting value. In geometric series it occurs precisely when the common ratio lies strictly between −1 and 1.

Expanding (a + b)ⁿ using coefficients from the binomial theorem, where each term involves a combination. Valid for positive integer n, and for any n when the expansion is infinite.

A unit of angle measure where one radian is the angle subtended at a circle's centre by an arc equal in length to the radius. π radians equal 180 degrees.

The distance along a curved portion of a circle's circumference. With angle θ in radians and radius r, it is given by s = rθ.

The area of the region bounded by two radii and an arc. With θ in radians it equals ½r²θ, which is a fraction θ/2π of the full circle.

An equation true for every value of the variable, such as sin²θ + cos²θ = 1. Identities are used to simplify expressions and prove further results.

Expressions for trigonometric functions of sums and differences, for example sin(A + B) = sin A cos B + cos A sin B. They underpin double-angle and R-form results.

Writing a sin θ + b cos θ as R sin(θ + α) or R cos(θ − α), where R = √(a² + b²). It helps find maximum and minimum values and solve equations.

The instantaneous rate of change of a function, found as the limit of the gradient of a chord as its length shrinks. It gives the gradient of the tangent at a point.

A rule for differentiating a function of a function: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. Essential for composite expressions.

A rule for differentiating a product: if y = uv then dy/dx = u dv/dx + v du/dx. It applies where two functions of x are multiplied together.

A rule for differentiating a fraction: if y = u/v then dy/dx = (v du/dx − u dv/dx)/v². It is used when one function divides another.

A point on a curve where the gradient is zero. It may be a maximum, a minimum or a point of inflection, classified using the second derivative or a sign test.

The derivative of the derivative, written d²y/dx². Its sign at a stationary point distinguishes a maximum (negative) from a minimum (positive).

The integral of a function between two limits, evaluated as the difference of an antiderivative at those limits. It gives the signed area under a curve.

The region between a curve and the x-axis, found by integrating the function between two limits. Parts below the axis give negative values, so split the interval carefully.

A technique that reverses the chain rule by replacing part of the integrand with a new variable. The differential and limits must be changed to match the substitution.

A vector giving a point's location relative to a fixed origin, often written with bold letters or arrow notation. Subtracting two position vectors gives the vector between the points.

A vector of length one, obtained by dividing any vector by its magnitude. It records direction only and is often written with a circumflex.

The product a·b = |a||b| cos θ, giving a single number. It tests perpendicularity (zero result) and finds the angle between two vectors.

For a 2 × 2 matrix with entries a, b, c, d, the value ad − bc. A non-zero determinant means the matrix is invertible and the transformation preserves or reverses area by that factor.

The matrix A⁻¹ that satisfies AA⁻¹ = I, the identity. For 2 × 2 matrices it equals (1/det A) times the matrix with a and d swapped and b, c negated.

A square matrix whose determinant is zero, so it has no inverse. Geometrically it squashes the plane onto a line or point, losing information.

The mean is the arithmetic average of a data set; the standard deviation measures typical spread about that mean. Together they summarise centre and dispersion.

The average of the squared deviations from the mean, equal to the square of the standard deviation. Squaring makes it sensitive to outliers and keeps it non-negative.

A symmetric bell-shaped continuous distribution defined by its mean and standard deviation. About 68% of values lie within one standard deviation of the mean.

An arrangement of objects where order matters, counted by nPr = n!/(n − r)!. It differs from a combination, where order is irrelevant.

A selection of objects where order does not matter, counted by nCr = n!/(r!(n − r)!). It appears in the binomial theorem and in probability calculations.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B). It applies when the sample space is restricted by known information.

Two events where the occurrence of one does not change the probability of the other, so P(A ∩ B) = P(A)P(B). This is not the same as mutually exclusive events.