CAPE Pure Maths Unit 1 key terms

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

Rewriting ax² + bx + c in the form a(x + h)² + k by adding and subtracting a constant. This reveals the vertex of the parabola and the maximum or minimum value.

When a polynomial f(x) is divided by (x − a), the remainder equals f(a). Dividing by (ax − b) leaves remainder f(b/a).

A number a is a root of polynomial f(x) exactly when (x − a) is a factor of f(x). It follows directly from the remainder theorem with zero remainder.

An expression formed by adding terms of the form axⁿ where n is a non-negative integer and a is a constant. Its degree is the highest power present.

Splitting a single rational expression into a sum of simpler fractions whose denominators are the factors of the original denominator. Useful for integration and for series expansion.

A rational expression in which the degree of the numerator is greater than or equal to the degree of the denominator. It must be divided out before partial fractions are formed.

A value of the variable that makes the equation true. For f(x) = 0 the roots are the x-intercepts of the graph of y = f(x).

A root that occurs more than once in a polynomial equation, so its factor appears with a power above one. The graph touches the x-axis there instead of crossing it.

A statement comparing two expressions with symbols such as <, ≤, > or ≥. Its solution is a range of values, often written in interval notation or shown on a number line.

The absolute value of a number, written |x|, giving its distance from zero and so always non-negative. The graph of y = |f(x)| reflects any part below the x-axis upward.

A rule that assigns to each element of a domain exactly one element of a codomain. Passing a vertical line test confirms that a graph represents a function.

The complete set of input values for which a function is defined. Values causing division by zero or a negative square root must be excluded.

The set of all output values a function actually produces as its input runs through the domain. It is read from the vertical spread of the graph.

A function in which different inputs always give different outputs, so its graph passes the horizontal line test. Only such functions possess an inverse that is also a function.

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.

A function built by applying one function and then another, written fg(x) or f(g(x)). The domain of the inner function restricts the composite.

Changing a graph by translation, stretch or reflection. Replacing x by x − a shifts right by a, while af(x) stretches vertically by factor a.

The turning point of a parabola, where it reaches its maximum or minimum value. For y = a(x − h)² + k the vertex is at (h, k).

A straight line that a curve approaches ever more closely without meeting, as the variable tends to infinity or to a value where the function is undefined.

A function of the form aˣ with a positive base a not equal to one. Its graph always passes through (0, 1) and has the x-axis as an asymptote.

The power to which a base must be raised to yield a given number, so logₐ y = x means aˣ = y. Logarithms turn multiplication into addition.

A logarithm taken to base e, written ln x, where e is approximately 2.718. It is the inverse of the exponential function eˣ.

The steepness of a line, measured as the change in y divided by the change in x between two points. Parallel lines share the same gradient.

Two lines meeting at a right angle. Their gradients m₁ and m₂ satisfy m₁m₂ = −1, so each gradient is the negative reciprocal of the other.

The point exactly halfway between two given points. Its coordinates are the averages of the corresponding x-coordinates and y-coordinates.

The set of points a fixed distance from a centre. With centre (a, b) and radius r its equation is (x − a)² + (y − b)² = r².

The angle subtended at the centre of a circle by an arc equal in length to the radius. One full turn equals 2π radians, or 360 degrees.

The distance along a curved part of a circle's circumference. For radius r and angle θ in radians it equals rθ.

The region of a circle enclosed by two radii and an arc. Its area is ½r²θ when θ is measured in radians.

An equation true for every value of the variable, such as sin²θ + cos²θ = 1. It can be used to simplify expressions but is not solved for specific values.

A rule expressing a trigonometric ratio of a sum or difference in terms of ratios of the separate angles, for example sin(A + B) = sin A cos B + cos A sin B.

A special case of the compound angle formulas with A = B, giving results such as sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A.

The value a function approaches as its input approaches a particular point, whether or not the function is defined there. Limits underpin the definition of the derivative.

The instantaneous rate of change of a function, found as the limit of a gradient between nearby points. Geometrically it gives the gradient of the tangent to the curve.

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

The rule for differentiating a product: if y = uv then dy/dx = u dv/dx + v du/dx, where u and v are both functions of x.

The rule for differentiating a fraction: if y = u/v then dy/dx = (v du/dx − u dv/dx)/v², with v not equal to zero.

A straight line touching a curve at one point and having the same gradient as the curve there. Its gradient equals the derivative at that point.

The straight line through a point on a curve that is perpendicular to the tangent there. Its gradient is the negative reciprocal of the curve's gradient at that point.

A point on a curve where the gradient is zero, so the tangent is horizontal. It may be a maximum, a minimum or a point of inflection.

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

A point where a function reaches the highest value in its neighbourhood. At such a stationary point the second derivative is negative.

A point where a function reaches the lowest value in its neighbourhood. At such a stationary point the second derivative is positive.

A point where the curve changes between concave upward and concave downward. The second derivative is zero there, though the gradient need not be zero.

A function whose derivative is the given function. Every continuous function has infinitely many, differing only by an added constant.

The family of all antiderivatives of a function, written with a constant of integration. It represents the reverse process of differentiation.

The limit of a sum of areas under a curve between two limits, written with lower and upper values. It produces a number, not a family of functions.

The region between a graph and the x-axis over an interval, found by integrating the function. Parts below the axis contribute negative values, so signs must be handled carefully.

The region enclosed by two graphs, found by integrating the upper function minus the lower one between their points of intersection.

The solid formed by rotating a region about an axis. Its volume is found by integrating πy² with respect to x, or πx² with respect to y.

A numerical method estimating a definite integral by treating the region as a series of trapezia. Accuracy improves as the number of strips increases.

How quickly one quantity changes with respect to another, usually time. It is found by differentiating, and related rates link quantities through the chain rule.

Finding the greatest or least value a quantity can take, often subject to a constraint. It is solved by forming an equation, differentiating and testing stationary points.