direct proportion
If two quantities are in direct proportion, as one increases, the other increases by the same percentage.
If y is directly proportional to x, this can be written as y ∝ x

indirect proportion
Inverse proportion is when one value increases as the other value decreases.

Straight line
y ∝ x
y = kx
This equation gives us a straight line. The gradient of the line is k.

Quadratic
y ∝ x2
y = kx2
This equation gives us a curve. The larger the value of k, the steeper the graph.

Cubic
y ∝ x3
y = kx3

Square root
y ∝square root of x
y = ksquare root of x
You have similar shaped curves for any powers between 0 and 1. Again, increasing k will make the graph steeper.

Inverse proportion
y ∝ 1/x ²
y = k/x ²
Inverse proportion leads to curved graphs.

surds
square root of ab = square root of a x square root of b
square root of a x square root of a = a

equations with fractions

graph transformations
f(x) + a [0/a]
f(+ ) = left f(x- ) = right
2f(x) = y coordinates x 2
-f(x) = reflection of f(x)
f(2x) = 1/2 x coordinate

sin cos tan graphs

factorising

quadratic equations
To solve a quadratic equation, the first step is to write it in the form: ax2 + bx + c = 0. Then factorise the equation

completing the square
This is another way to solve a quadratic equation if the equation will not factorise.
It is often convenient to write an algebraic expression as a square plus another term. The other term is found by dividing the coefficient of x by 2, and squaring it.

quadratic formula
Solve 2x2 - 5x - 6 = 0
Here a = 2, b = -5, c = -6

the quadratic formula

A taxi firm charges £0.50 per mile plus a fixed charge of £2.00. Write down a formula for the cost (C) of hiring this taxi to travel 'n' miles.

Collecting like terms
To simplify an expression, we collect like terms.
Look at the expression: 4x + 5x -2 - 2x + 7

(2x + 5) squared

multiplying out brackets
Multiply out: 3(4x - 7)

multiplying out negatives
Multiply out the expression:
-3(2n - 8)

multiplying out 2 brackets
Multiply out these two brackets:
(x + 4) (x + 3)

FOIL
first outer inner last

proof
prove
(3n + 1) squared - (3n - 1) squared
is a multiple of 6

Index notation:
used to represent powers

index multiplying and dividing

solving equations

solving inverse equations
Here is how to solve 2a + 3 = 7 using inverses.

unknowns on both sides of the equation
Solve the equation 3b + 4 = b + 12, and find the value of b.

equations with brackets
Solve the equation: 3(b + 2) = 15

trial and improvement
Find the answer to the equation x3 – 2x = 25 to one decimal place.

simultaneous equations

solving inequalities
Solve the expression 3x - 7 < 8

double inequalities
Solve the inequality -4 < 2x - 6 < 12

equations for a straight line graph can contain

tables for graph plotting

On a graph, parallel lines have the same gradient.
For example, y = 2x + 3 and y = 2x - 4 are parallel because they both have a gradient of 2.

the lines y = 2x + 3 and y = -1/2 x -1 cross at right angles

y = mx + c
m = gradient
c = y intercept

cubic graphs

Calculate the points and then plot the graph for the equation y = x3 - x + 8

reciprocal graphs
Graphs of the form y = 1/x, 2/x, 3/x

quadratic sequences
If the difference between the terms changes, this is called a quadratic sequence.
If you use the formula n^2 + n to make a sequence, it means that:
When n = 1 you get 1^2 + 1 = 2
When n = 2 you get 2^2 + 2 = 6
When n = 3 you get 3^2 + 3 = 12
When n = 4 you get 4^2 + 4 = 20
- giving the sequence 2, 6, 12, 20.

histograms
frequency density = frequency / class width

frequency tables
finding the mean and median

frequency polygons
dot in midpoint

median upper quartile lower quartile and interquartile range of frequency

stem and leaf diagram

box plots

Amit is 12 years old. His brother, Arun, is 9.
Their grandfather gives them £140, which is to be divided between them in the ratio of their ages. How much does each of them get?

standard form

adding and subtracting standard form

multiplying and dividing standard form

parts of a circle

circumference of a circle

area of a circle

area of a semi circle

perimeter of a semi circle

circle theorems

angle at the centre of a circle

angles in the same segment are equal

angles in a semi circle are 90 degrees
The angle at the centre (AOB) is twice the angle at the circumference (APB). As AOB is 180°, it follows that APB is 90°

Opposite angles in a cyclic quadrilateral add up to 180°
A cyclic quadrilateral is a quadrilateral whose vertices all touch the circumference of a circle. The opposite angles add up to 180o.
In the cyclic quadrilateral below, angles A + C = 180o, and angles B + D = 180o

The angle between the tangent and radius is 90°
A tangent to a circle is a line which just touches the circle.

difficult circle theorem

quadrilateral has been divided into two triangles, so the interior angles add up to 2 × 180 = 360°.

interior angles of polygons

The volume of a pyramid is:
1/3 × base area × perpendicular height
The surface area of a pyramid is the area of the base + the area of all the other sides.

The volume of a cone is:
1/3πr^2h
The curved surface area of a cone is:
πrL
(where L is the slant height).
Note this doesn't include the area of the base, which is a circle, area πr^2.

The volume of a sphere is
4/3πr^3
The surface area of a sphere is:
4πr^2

volume and area

3 figure bearings

Pythagoras

the sine rule

the cosine rule

area of a triangle