Amplitude
The maximum displacement from the principal axis in a trigonometric function. For \(y = a\sin(bx) + c\), amplitude = \(|a|\).
Arithmetic Progression (AP)
A sequence with a constant difference between consecutive terms.
Binomial Coefficient
The coefficient of a term in the expansion of \((a+b)^n\), given by \(\binom{n}{r} = \frac{n!}{(n-r)!r!}\).
Chain Rule
A rule for differentiating composite functions: \(\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}\).
Combination
A selection of objects where order does not matter. \(\binom{n}{r} = \frac{n!}{(n-r)!r!}\).
Common Difference (\(d\))
The constant difference between consecutive terms in an arithmetic progression.
Common Ratio (\(r\))
The constant ratio between consecutive terms in a geometric progression.
Completing the Square
A method of rewriting a quadratic expression in the form \(a(x-h)^2 + k\).
Composite Function
A function formed by applying one function and then another: \(fg(x) = f(g(x))\).
Convergent Series
A geometric series with \(|r| < 1\) that has a finite sum to infinity.
Derivative
The rate of change of a function; the gradient of the tangent to a curve.
Discriminant
For \(ax^2 + bx + c = 0\), \(\Delta = b^2 - 4ac\). Determines the nature of roots.
Domain
The set of all possible input values (\(x\)-values) for a function.
Factor Theorem
\((x-a)\) is a factor of \(P(x)\) iff \(P(a) = 0\).
Factorial
\(n! = n \times (n-1) \times \cdots \times 1\), with \(0! = 1\).
Function
A rule that maps each input to exactly one output.
Geometric Progression (GP)
A sequence with a constant ratio between consecutive terms.
Gradient
The slope of a line, \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Inverse Function
A function that "undoes" the effect of another function. \(f^{-1}(f(x)) = x\).
Kinematics
The study of motion of a particle in a straight line.
Linearisation
Converting a non-linear relationship to a straight line form by taking logarithms.
Logarithm
The exponent to which a base must be raised to get a number: \(\log_a y = x \iff a^x = y\).
Midpoint
The point halfway between two points: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
Modulus
The absolute value of a number: \(|x| = x\) if \(x \ge 0\), \(-x\) if \(x < 0\).
Normal
A line perpendicular to the tangent at a point on a curve.
One-One Function
A function where each output comes from exactly one input: \(f(a) = f(b) \Rightarrow a = b\).
Period
The interval after which a trigonometric function repeats its values.
Permutation
An arrangement of objects where order matters. \(^nP_r = \frac{n!}{(n-r)!}\).
Perpendicular Bisector
A line perpendicular to a segment that passes through its midpoint.
Position Vector
A vector that starts at the origin.
Principal Axis
The horizontal line about which a trigonometric function oscillates; for \(y = a\sin(bx) + c\), the principal axis is \(y = c\).
Product Rule
\(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}\).
Quadratic
A function of the form \(f(x) = ax^2 + bx + c\) (\(a \neq 0\)).
Quotient Rule
\(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\).
Radian
An angle measure; 1 radian \(= \frac{180^\circ}{\pi} \approx 57.3^\circ\). \(s = r\theta\) and \(A = \frac{1}{2}r^2\theta\) require radians.
Range
The set of all possible output values (\(y\)-values) for a function.
Remainder Theorem
When \(P(x)\) is divided by \((x-a)\), the remainder is \(P(a)\).
Resolving Vectors
Breaking a vector into horizontal and vertical components: \(v_x = v\cos\theta\), \(v_y = v\sin\theta\).
Scalar
A quantity with magnitude only (e.g., speed, mass).
Sector
A region of a circle bounded by two radii and an arc.
Stationary Point
A point on a curve where \(f'(x) = 0\) (maximum, minimum, or point of inflection).
Tangent
A line that touches a curve at exactly one point; the gradient of the tangent is \(f'(x)\).
Unit Vector
A vector with magnitude 1.
Vector
A quantity with both magnitude and direction (e.g., displacement, velocity, force).