Unit 1

The Straight Line

Gradients

m = tanθ

Distance Formula

Midpoint Formula

Parallel lines

Perpendicular lines

y = mx + c

y - b = m( x - a)

Ax + Bx + Cx = 0

Collinearity

Intersection of straight lines

Perpendicular bisectors

Altitudes of  a triangle

Centroids, circumcentres

Orthocentre

Medians of a triangle

Functions and Graphs

Set notation

Domains

codomains

range

image

function

composite functions

inverse functions

exponential functions

logarithmic functions

standard graphs

graph of y = f(x) + a

graph of y = f(x+ a)

graph of y =  - f(x)

graph of y =  f(-x)

graph of y =  kf(x)

graph of y =  f(kx)

exponential graphs

logarithmic graphs

Period & amplitude

trigonometric graphs

radians

exact values

graph of y = acos(nx)

graph of y = asinx

graph of y = sin(ax+ b)

graph of y = cos(ax+ b)

graph of y = (1-sinx)² +2

Differentiation

gradient of tangent to curve

notation  dy/dx   and   f'(x)

if f(x) = xⁿ , f '(x) = nx^n-1

if f(x) =g(x) + h(x) , f '(x) = g'(x) = h'(x)

if f(x) = kg(x), f'(x) = kg'(x)   k is a constant

stationary points

maximum / minimum turning points

horizontal point of inflexion

Recurrence relations

nth term of sequence

linear recurrences  u_n+1=mu_n+c

Convergence / Divergence

Calculating limits   L=c/(1-m) 

Arithmetic sequence    u_n+1=u_n+b

Geometric sequence    u_n+1=au_n

Fibonacci  sequences   u_n+2=u_n+1+u_n

Unit 2

Polynomials and Quadratic Theory

Polynomials

Nested method

Synthetic division

Remainder theorem

Factor theorem

Solving polynomial equations

Approximate roots of f(x)

Solving quadratics by :-

….Factorisation

….Completing the square

….Quadratic formula

….graphical means

Tangents to curves

Discriminant

solving quadratic inequalities

Integration

Anti -differentiation

Differential equations

Integration using formula

Area under a curve

Definite integrals

Area between  two curves

The Circle

x²+y²=r²

centre (a,b)   (x-a)²+(y-b)²=r^{2  ,}

General equation x²+y² +2gx +2fy+c=0

Tangents to a circle

Intersections of lines and circles

Trigonometry and Compound Angle Formulae

Revision basics, sine rule, cosine rule, area of triangle.

3D trigonometry

3D  co-ordinates

Compound angles:-

cos(A+B) cos(A - B)

sin(A+B)  sin(A - B)

Sin2A  Cos2A

Trigonometric equations

Further trig equations

Graphs y=sin(x+α) , y=sin(x-α)

Graphs y=cos(x+α) , y=cos(x-α)

Unit 3

Vectors

Scalars and vectors

vectors and directed line segments

Magnitude of vectors

Addition, subtraction of vectors

Multiplicatipon by a scalar

Position vector

Section formula

Unit vectors

Scalar product  a.b

Distributive law for scalar product

Further Calculus

Differentiate trig functions

Chain rule for differentiation

Standard integrals

Integrate trig functions

Area under trig curves

Exponential and logarithmic Functions

exponential growth and decay

the number e

y=a^x   x = log_ay

Log laws

solving exponential equations

Using graphs to solve y = axⁿ or y=ab^x

The Wave Function

Rcos(x-α)

Rcos(x+α)

Rsin(x-α)

Rsin(x+α)

Maxima and minima

Trig equations