Understand the terms: function, domain, range (image set), one-one function, many-one function, inverse function and composition of functions [includes explaining in words why a given function is a function]
Find the domain and range of functions [includes inverse functions and composite functions]
Recognise and use function notation
[f(x) = 2ex, f:x ↦ lg x, for x > 0, f-1(x), fg(x) = f(g(x)), f2(x) = f(f(x))]
Understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic, cubic or trigonometric. If f(x) is trigonometric it will be one of the following:
[y = a sin bx + c, y = a cos bx + c, y = a tan bx + c]
where a is a positive integer, b is a simple fraction or integer, and c is an integer. Fractions will have a denominator of 2, 3, 4, 6 or 8 only.
Explain in words why a given function does not have an inverse.
Find the inverse of a one-one function.
Form and use composite functions.
Use sketch graphs to show the relationship between a function and its inverse [understand that each function is the reflection of the other in the line y = x]
[2] Quadratic Functions
Find the maximum or minimum value of the quadratic function f:x ↦ ax2 + bx + c by completing the square or by differentiation.
Use the maximum or minimum value of f(x) to sketch the graph of y = f(x) or determine the range for a given domain.
Know the conditions for f(x) = 0 to have:
two real roots
two equal roots
no real roots
and the related conditions for a given line to:
intersect a given curve
be a tangent to a given curve
not intersect a given curve
Understand how the discriminant relates to the roots of the equation
Solve quadratic equations for real roots.
Find the solution set for quadratic inequalities either graphically or algebraically.
[3] Factors of Polynomials
Know and use the remainder and factor theorems
Find factors of polynomials
Solve cubic equations
[4] Equations, Inequalities and Graphs
Solve equations of the type
|ax + b| = c (c \(\geq\) 0)
|ax + b| = cx + d
|ax + b| = |cx + d|
|ax2 + bx + c| = d
Using algebraic or graphical methods
Solve graphically or algebraically inequalities of the type
k|ax + b| > c (c \(\geq\) 0)
k|ax + b| \(\leq\) c (c > 0)
k|ax + b| \(\leq\) |cx + d|
where k > 0
|ax + b| \(\leq\) cx + d
|ax2 + bx + c| > d
|ax2 + bx + c| \(\leq\) d
Use substitution to form and solve a quadratic equation in order to solve a related equation.
Sketch the graphs of cubic polynomials and their moduli, when given as a product of three linear factors.
Solve graphically cubic inequalities of the form
f(x) \(\geq\) d, f(x) > d, f(x) \(\leq\) d, f(x) < d
where f(x) is a product of three linear factors and d is a constant
[5] Simultaneous Equations
Solve simultaneous equations in two unknowns by elimination or substitution
[6] Logarithmic and Exponential Functions
Know and use simple properties and graphs of the logarithmic and exponential functions, including ln x and ex.
Logarithms may be given to any base.
Understand that f(x) = ex and g(x) = ln x are each the inverse of the other.
Understand the asymptotic nature of the graphs of logarithmic and exponential functions.
State the equations of any asymptotes.
Graphs are limited to y = kenx + a and y = k ln(ax + b) where n, k, a and b are integers.
Know and use the laws of logarithms, including change of base of logarithms.
Solve equations of the form ax = b
[7] Straight-Line Graphs
Use the equation of a straight line.
Know and use the condition for two lines to be parallel or perpendicular.
Solve problems involving midpoint and length of a line, including finding and using the equation of a perpendicular bisector.
Transform given relationships to and from straight-line form, including determining unknown constants by calculating the gradient or intercept of the transformed graph.
[8] Coordinate Geometry of the Circle
Know and use the equation of a circle with radius r and centre (a, b).
(x - a)2 + (y - b)2 = r2
x2 + y2 + 2gx + 2fy + c = 0
Solve problems involving the intersection of a circle and a straight line.
Includes determining whether a straight line:
is a tangent
is a chord
does not intersect the circle
Solve problems involving tangents to a circle. Includes finding equations of tangents.
Solve problems involving the intersection of two circles. Includes finding points of intersection, finding the equation of a common chord or determining whether two circles:
intersect
touch
do not intersect
[9] Circular Measure
Solve problems involving the arc length and sector area of a circle, including knowledge and use of radian measure.
[10] Trigonometry
Know and use the six trigonometric functions of angles of any magnitude [sine, cosine, tangent, secant, cosecant, cotangent]
Understand and use the amplitude and period of a trigonometric function, including the relationship between graphs of related trigonometric functions.
Draw and use the graphs of
y = a sin bx + c
y = a cos bx + c
y = a tan bx + c
where a is a positive integer, b is a simple fraction or integer (fractions will have a denominator of 2, 3, 4, 6 or 8 only), and c is an integer
Solve, for a given domain, trigonometric equations involving the six trigonometric functions.
Prove trigonometric relationships involving the six trigonometric functions.
[11] Permutations and Combinations
Recognise the difference between permutations and combinations and know when each should be used.
Know and use the notation n! and the expressions for permutations and combinations of n items taken r at a time [0! = 1]
Solve problems on arrangement and selection using permutations or combinations.
[12] Series
Use the binomial theorem for expansion of (a + b)n for positive integer n.
Use the general term nCr an-rbr, 0 \(\leq\) r \(\leq\) n (knowledge of the greatest term and properties of the coefficients is not required)
Recognise arithmetic and geometric progressions and understand the difference between them.
Use the formulas for the nth term and for 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 [includes explaining why a particular geometric progression has or does not have a sum to infinity]
[13] Vectors in Two Dimensions
Use vectors in any form, e.g. \(\begin{pmatrix}
a \\
b
\end{pmatrix}\), \(\overrightarrow{AB}\) and ai - bj
Know and use position vectors and unit vectors
Find the magnitude of a vector; add and subtract vectors and multiply vectors by scalars.
Compose and resolve velocities.
Determine a resultant vector by adding two or more vectors together.
Includes the use of a velocity vector to determine position and solve problems in context such as particles colliding.
[14] Calculus
Understand the idea of a derived function [only an informal understanding of the idea of a limit is expected, and the technique of differentiation from first principles is not required]
Use the notations f'(x), f"(x), \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\)
Use the derivatives of the standard functions xn (for any rational n), sin x, cos x, tan x, ex, ln x, together with constant multiples, sums and composite functions (use of chain rule). For trigonometric functions angles will always be in radians.
Differentiate products and quotients of functions.
Apply differentiation to find gradients, tangents and normals, stationary points, connected rates of change, small increments and approximations and practical maxima and minima problems.
Use the first and second derivative tests to discriminate between maxima and minima.
Understand integration as the reverse process of differentiation. Solutions for indefinite integrals should include an arbitrary constant.
Integrate sums of terms in powers of x including \(\frac{1}{x}\) and \(\frac{1}{ax + b}\)
Integrate functions of the form (ax + b)n for any rational n, sin(ax + b), cos(ax + b), sec2(ax + b), eax + b
Evaluate definite integrals and apply integration to the evaluation of plane areas. Plane areas include: between a line and a curve, between two curves, a sum of two areas.
Apply differentiation and integration to kinematics problems that involve displacement, velocity and acceleration of a particle moving in a straight line with variable or constant acceleration.
Make use of the relationships to draw and use the following graphs:
