Completing the square (locate the vertex and sketch the graph).
Find the discriminant of a quadratic polynomial (determine the number of real roots).
Solve quadratic equations and quadratic inequalities in one unknown.
Solve by substitution simultaneous equations where one is linear and one is quadratic.
Functions
Understand function terms: domain, range, one-one function, inverse function, and composite of functions.
Identify the range of a function.
Find the composite of two functions, ensuring the range of the first is within the domain of the second.
Determine if a function is one-one and find its inverse.
Graphically illustrate the relation between a one-one function and its inverse (include a mirror line y = x).
Use transformations of the graph of y = f(x) such as y = f(x) + a, y = f(x + a), y = af(x), y = f(ax), and combinations (use 'translation', 'reflection', and 'stretch').
Coordinate Geometry
Find the equation of a straight line given two points or one point and the gradient.
Use forms y = mx + c, y - y1 = m(x - x1), ax + by + c = 0 to solve problems (including distances, gradients, midpoints, points of intersection, and the relationship between gradients of parallel and perpendicular lines).
Equation (x - a)2 + (y - b)2 = r2 represents a circle with center (a, b) and radius r (including the expanded form x2 + y2 + 2gx + 2fy + c = 0).
Solve problems involving lines and circles (including properties of circles such as tangent perpendicular to radius, angle in a semicircle, symmetry).
Use the relationship between points of intersection of graphs and solutions of equations (e.g., find the set of values of k for which the line y = x + k intersects, touches, or does not meet a quadratic curve).
Circular Measure
Use the relationship between degrees and radians.
Use the formulae s = rθ and \(\frac{1}{2}\) r2θ for arc length and sector area of a circle.
Calculate lengths and angles in triangles and areas of triangles.
Trigonometry
Sketch and use graphs of sine, cosine, and tangent functions (angles in degrees or radians).
Use the exact values of sine, cosine, and tangent for 30°, 45°, 60°.
Use the notations sin-1x, cos-1x, tan-1x for the principal values of the inverse trigonometric relations.
Use the identities tanθ = \(\frac{sinθ}{cosθ} \) and sin2θ + cos2θ = 1.
Find all solutions of trigonometric equations within a specified interval.
Series
Use the expansion of (a + b)n, where n is a positive integer.
Recognize arithmetic and geometric progressions.
Use the formulae for the nth term and the sum of the first n terms to solve problems involving arithmetic or geometric progressions.
Use the condition for the convergence of a geometric progression and the formula for the sum to infinity of a convergent geometric progression.
Differentiation
Understand the gradient of a curve at a point, using notations f'(x), f"(x), \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) for first and second derivatives.
Use the derivative of xn (for any rational n), with constant multiples, sums and differences of functions, and composite functions using the chain rule.
Apply differentiation to gradients, tangents and normals, increasing and decreasing functions, and rates of change.
Locate stationary points and determine their nature.
Integration
Understand integration as the reverse process of differentiation.
Integrate (ax + b)n (for any rational n except -1), along with constant multiples, sums and differences.
Evaluate definite integrals.
Use definite integration to find:
The area of a region bounded by a curve and lines parallel to the axes, or between a curve and a line, or between two curves.
The volume of revolution about one of the axes (may involve a region not bounded by the axis of rotation).
