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Algebra and Functions

Indices
Indices are numbers that are raised to a power (eg. 251/2). The laws of indices are as follows:


ya × yb = ya+b
ya ÷ yb= ya-b
y -b = 1/yb
ym/n = (ny)m
(yn)m = ynm
y0 = 1




Surds
Surds are numbers left in…

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In general


Fractions in the form multiply top and bottom by


Fractions in the form multiply the top and bottom by


Fractions in the form multiply the top and bottom by




Quadratic Functions
A quadratic equation is an equation where the highest power of x is x2, so it is…

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If b2 - 4ac = 0 then the quadratic formula says that x = - b/2a, so there is only
one solution. The graph will only touch the x-axis at one point, therefore.

However, if b2 - 4ac > 0, there will be 2 solutions to the equation and so…

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Example
Suppose we wish to fi
nd the minimum value of the quadratic function f(x) = x2-6x -12.
We know that because f(x) = x2-6x-12 has a positive x2 term the graph will have a minimum
value.
This will occur when (x -3)2 is zero. The minimum value will be…

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Example: Solve the simultaneous equations y - 2x = 1 and 2y - 3x = 5
Rearranging Equation 1, we get y = 1 + 2x. We can replace the `y' in equation 2 by
substituting it with 1 + 2x. Equation 2 becomes: 2(1 + 2x) - 3x =…

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Graphs and their Fuctions
Graphs are used to plot two variables (x and y) against each other. Simple graphs include
linear, quadratic, cubic and inverse.




Linear graphs are given in the form y=mx+c, where 'm' is
the gradient of the graph and 'c' is the intercept.




Quadratic graphs have x2…

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Graphs can be translated left and right by adding or subtracting quantities, and
stretched or squashed by multiplying or dividing by quantities.

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Co-ordinate Geometry

Distance Between Two Points:

The length of the line joining the points (x1, y1) and (x2, y2) is:




Example
Find the distance between the points (5, 3) and (1, 4).
(So in this case, x2 = 1, x1 = 5, y2 = 4 and y1 = 3).
Distance…

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Parallel and Perpendicular Lines:

If two lines are parallel, then they have the same gradient.

If two lines are perpendicular, then the gradients of the two lines are reciprocals of each other.



Example
a) y = 2x + 1
b) y = -½ x + 2
c) ½y = x…

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Recurrence Relationships:
This is where the next term of a sequence is defined using the previous term(s). To define a
recurrence relation, you have to give the first term. Then you give a formula to tell you how to work
out the next term from the previous ones.

For example,…

Comments

C Clarke

Very clear and helpful.  

katie

thank you ever so much :) this is really helpful

leah

no intergration or differentiation?

AruntujhaPonniah

Useful website

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