1. Straight from My Revision Book By the way, this is probably repeating the other one a lot (Special thanks to Moji who is a star )

2.

Transformations of Graphs

Congruency

Completing the square

Calculus

• Differentiation

• Integrals

• Finding the constant

• Stationary points

• Motion with Variable Acceleration

Binomial

• Distribution

• Expansion

The Discriminate

• What is the Discriminate

• Using the Discriminate

• Finding Minimum or Maximum Value

3. 4.

When we have to stretch a line along the x-axis, we have to stretch to nwhere n is the number given

• For example:

• • y=sin2x becomes y=sinx

When stretching/compressing a line along the y-axis, the stretch is to the number given

• For example

• • y=3sinx becomes y=3sinx

For example:

• Curve y=x translated to give y=f(x)

Move right first f(x-2)

• It is -2 because x is squared in the question and moved to the right

Move down -4

Therefore: y=(x-2)-4

Again it is squared because the question has x

y x 0 x A (2,-4) y=x 5.

Similarity : when the angles are the same but thelengths are different

Triangles are congruent if:

• 3 sides are the same (SSS)

• 2sides + included angle are the same (SAS)

• 2angles + 1side are the same (ASA or AAS)

• In a right angled triangle, the hypotenuse +1side are the same (RHS)

You will always be given the graph and an equation

For example (p408 ex17g)

• 2a) x+4x-5 first draw graph in book

• To solve the equation make a grid

• • • • =0 therefore y=0 when x=-5 and 1

Note : if this doesnt work y=0

x -6 -5 -4 -3 -2 -1 0 1 2 x 36 25 16 9 4 1 0 1 4 +4x -24 -20 -16 -12 -8 -4 0 4 8 -5 -5 -5 -5 -5 -5 -5 -5 -5 -5 y 7 0 -5 -8 -9 -8 -5 0 7 6.

When you are told to complete the square:

• Take the co-efficient of x at be, for example in:

• abc

• • x + 6x + 1

• Half this and then square it. Both add and subtract it from the equation (so the equation doesnt change) like so:

• • x + 6x + (6/2) (6/2) + 1

• This can be factorised into:

• • (x+3) + 8

• To check you can expand to get:

• • x +6x + 9 + (-9 +1)

• Remember: that if it is 4x + ... it has to be factorised into 4(x + ...) rather than (4x + ...)

7. 8.

Differentiation

Integrals

Finding the constant

Stationary points

Kinematics

9.

To differentiate:

• E.g. y=3x+4x-3x+2

For long equations, differentiate in parts and then add/subtract as needed

• 1 st(3x) multiply the power by themultiple of x=15x

• Then minus 1 from the power=15x

• Repeat for all parts, remembering the signs:

• • = 15x+16x - 6x [+2] ...(continued)

d dx Multiple of x Power 10.

For a whole number i.e. The 2, it will =0

Therefore the answer is:

• 15x+16x - 6x

Remember:

• For things like 5x they become 5

• x = x not x

• x = x = 1

• x = x

...(continued)

11.

The second derivative tells us if a point is a maximum point, minimum point or a possible point of inflection

This is just differentiating what you already have differentiated

The third derivative tells us if a point is a point of inflection if it does not equal zero. If it does equal zero it is not possible at this level.

• It shouldnt equal zero but if it does just remember to putnot possible at this levelas there may be a mistake in the question

12.

Integration is the inverse process of differentiation

The symbol for this is

To integrate:

• E.g. y=3x+4x-3x+2

Do it in parts:

• ( 3x+4x-3x+2)

• Take the multiple, like 3, and divide it by the power + 1

• In this case you would get 3/6. You then take the x, x here, and add 1 to the power again, x

• You place this over the multiple in the fraction to give you: [(3x)/6]

• Continue like this, adding and subtracting according to the equation given

• For a constant, like 2, simply at x, thus 2 = 2x

• Then simplify if possible

• • The example above becomes 2x

• Then add c where c is the constant

13.

There are two formulae that need to be learnt:

Note:Finding the original function given its derivative is the same as finding the integral

1. 2. Where a is a constant 14.

For definite integrals, numbers are given

• E.g.

You must integrate as normal

Then you take the number for the upper limit and substitute it in for x, x=3

Do the same for the lower limit, x=1, and subtract this from the previous value. Giving:

Note:As c vanishes when you work the previous expression, it is common to replace c altogether and write:

It can also be found using the formula:

This can be used to find areas of curves

Upper limit Lower limit 15.

To find the constant, you will be given co-ordinates which then can be substituted into an expression which will be used to find the constant

• For example:

IntegrateSubstitute 16.

DONT FORGET TO PUT IT INTO THE PREVIOUS EXPRESSION!

• So the answer is:

You could lose marks for leaving it just like that!

17.

Explain whyis a possible expression for the gradient of the curve and give an alternative expression for

• For the first part:

• • • because

• • • So x=0 or x=2making them stationary points thus a possible expression

• For the second part:

• • Where k is a constant which is an alternative expression

The curve passes through the point (3,2). Taking as , find the equation of the curve

• Find the integral:

• Find the constant:

• Substitute this into the equation from the integration:

• • Or

18.

The 2 ndderivative tells us if a point is a maximum, a minimum, or a possible point of inflection

The 3 rdderivative tells us if a point is a point of inflection if it doesnt equal zero

If it does equal zero it is not possible at this level

However this should only occur if there was a mistake in workings or the question

19.

For

Find the first derivative

• So, or

Then find the 2 ndderivative

Substitute one possible value for x into this equation

• So at x=2 a minimum

• So at x=-1 a maximum

You will often be asked to find the possible points (co-ordinates) where the stationary points are. To do this, simply substitute the possible values of x into the equation given

Therefore:

• At x=2 is a minimum

• At x=-1 is a maximum

It is important to say that the co-ordinate is a max/min/point of inflection

20.

Key:

• v = velocity

• u = initial velocity

• a = acceleration

• t = time

• s = displacement

Equations:

• v = u + at

• s = [(u + v)/2] X t

• v = u + 2as

To find an equation for v you can do the derivative of s (ds/dt)

To find an equation for a you can do the derivative of v (dv/dt)

Example:

• Find an equation for v with the following equation

• s = 5t - t + 3

• ds/dt = 10t 1

• Thus v = 10t 1

• Find an equation for a with answer above

• dv/dt = 10 so a = 10

Similar to doing the 2 ndderivative

21.

v = ds/dts = v dt

a = dv/dtv = a dt

Area under the line which also given the velocity in a time-distance graph

Example:

• a = 2 6t(find v)

• v = a dt

• (2-6t) = 2t [(6t)/2] + c

• v = 2t 3t + c

• • At t = 0; v = 1

• 1 = 0 0 + c

• c = 1

• v = 2 3t + 1

• s = v dt

• (2 3t + 1 )

• • At t = 0; and s = 0

• = [(2t)/2] [(3t)/3] + t

• s = t - t + t + c

• 0 = 0 0 + 0 + c

• c = 0

22.

Distribution

Expansion

• Pascals Triangle

23.

Two mutually exclusive events (meaning that if something happens, something else cannot happen)

If a trial is conducted n times,

• P is the probability of success in every trial

• q is the probability of failure

• • Note:

The probability of exactly r successes is:

• Where x is number of success and r=0,1,2...

24.

Remember:

• Indices of a term = the sum of indices of expression

• • For example:

However, Pascals Triangle is long to use so thebutton on the calculator is used instead

So, if we want to work out the co-efficient ofin the expression, we need

Another way is:

In general this is:

Remember:

Using the co-efficients of each expansion we can get Pascals Triangle

Its easy to work out as you just add the previous numbers

• For example: 1211331

The triangle is symmetrical and always has 1 at the beginning and end of each line

Power of binomial Power of x and 25. 26.

What is the Discriminate

Using the Discriminate

Finding Minimum or Maximum Value

27.

Key: = discriminate

When using the equation the decides the answer you get

If, you get anon-realnumber as it is a minus

If , you gettworeal distinctanswers

If, you getone real equalanswer

It is important to know the correct terminology here

28.

For example:

• abc

• Useto find the discriminate so it will be:

• Then, when , you can factorise to get

• • So,

• • Or

Whenorthe answers are above the x-axis therefore the answer is:

29.

If given an equation where the constant is negative to find the minimum, the only possible answer is for the equation to equal zero

For example:

• Ascannot be any less than 0, when this is multiplied by 4 it can be no less than zero

• Therefore -25 has o be the minimum possible answer