Maths need to know - Owen Shepherd

Role of a in quadratic

Control the width and direction of quadratic

Higher = steeper

Role of b in a quadratic

Control the axis of symmetry

Where it is reflected

Role of c

Y intercept

Sampling methods

Census, Sampling

Census sampling sampling methods

Simple random sampling, systematic, stratified, quota

Sample sampling methods

Opportunity (Convenience sampling)

Simple random sampling

Every person allocated a number (1-n), random numbers selected, Draw numbers until sample size is met

Simple random sampling pros and cons

Pros: Not biased - random spread

Cons: Time consuming, whole population needed, difficult to set up and manage

Systematic sampling

The whole population is sampled at regualr intervals,

Number 1-n, select every 5th number etc, until sample size is met

Systematic sampling pros and cons

Pros: Easy to set up, likely to be unbiased

Cons: Whole population needed, if the sleection interval coincide with a pattern it is biased

Stratified sampling

Population divided into groups of similar characteristic, same proportion taken from each group

Groups are called strata

Group data, split into groups, fine relevant number from each group for sample size

Stratified sampling pros and cons

Pros: If categories are mutually exclusive gives a good representation, useful when results differ

Cons: The process to select who to select and in what proportion can be difficult and inefficient

Cluster sampling

Population divide into distinct groups, randomly select a group of people to sample

Cluster sampling

Pros: Can be easy, More practical

Cons: Risk of bias in the group, could be less representative

Opportunity / convenience sampling

Sample chosen from a section of the population mose convenient for the sampler

e.g. the first people they see

Opportunity / convenience sampling pros and cons

Pros: Data gathered easily and quickly

Cons: There is no method to remove bias, could be not representative

Quota sampling

Sampler is given a quota of data from each group of x they need to collect

e.g. 3 male and 2 female

Quota sampling pros and cons:

Pros: It is easy for the interviewer, access select groups

Cons: Can be biased, groups excluded

Volunteer sampling

People choose to be part of the sample

Volunteer sampling pros and cons

Pros: Little time and effort, encourage engagement

Cons: Can be a lot of bias, people volunteer with strong oppinions

Linear function form

Y = mx + c

M = gradient

C = intercept

Straight line, of gradient m passing through point (a,b) is?

Y - b = m(x - a)

Same gradient means

Parallel lines

Perpendicular if

Gradients multiply to -1

Midpoint equation?

(X1 + X2)/2 + (Y1 + Y2)/2

Perpendicular bisector is?

The line that passed perpendicularly thought the line of AB, at the Midpoint of AB

X axis is when

Y = 0

Y axis when

X = 0

Translated quadratic form

Y = (x - a)² + b

By vector (a/b)

How to find vertex of quadratic

Complete the square

Discriminant

b² - 4ac

Discriminant greater than 0

2 real roots

Discriminant = 0

One real root

Discriminant < 0

No roots

Quadratic equation 3 ways to solve

Factorise

Complete the square

Quadratic Formula

Surd rules

root a x root b = root ab

Square numbers cancel out into multiples

Rationalise the denominator by times by 1 = surd / surd

Rationalise denominator when sued in form (root a) - b

Times by (root a) + b

a^b x a^c =

a^{b + c}

a^b / a^c =

a^{b - c}

(a^b)^c =

a^bc

a^-b =

1/a^b

a^1/b =

root b a

a⁰ =

1

loga(b) + loga(c) =

loga(bc)

loga(b) - loga(c) =

loga(b/c)

c x loga(b) =

loga(b^c)

loga(a) =

1

loga(1) =

0

(-(x))² =

x²

Two pairs of equal roots means?

Solve the hidden quadratic

Use Discriminant for 1 root

Then, that root is the root for the + value

Other root is the negative of the value

a^x x b^{x =}

(ab)^x

How to divide / multiply polynomials with other polynomials?

Grid method

Factor theorem

If factor = (x-a)

Then f(a) = 0

Factor theorem 2

(ax - b)

f(b/a) = 0

If polynomial factored in to three brackets does it have three roots

Can but not always

What is it known as if = (x-3)(x-2)(x-2)

Repeated real root of x = 2

Describe: f(x-a) + b

Translation by vector (a b)

Describe: f(kx)

Stretched parallel to the x axis by Scale Factor 1/k

Describe: kf(x)

Stretched parallel to the y axis by Scale Factor k

Describe: f(-x)

Reflection in the y axis

Describe: -f(x)

Reflection in the x axis

Circle equation

(x-a)² + (y-b)² = r²

compare how a line intersects a circle?

line in form y = mx + c

Substitute into circle equation

Solve out to form quadratic = ax² + bx + c

Use the discriminant

Three Circle theorems to know

Tangent and radius at 90

Two tangents that meet at same point are equal in length - makes kite - see image

Three points, with a 90 degree angle are three points on the circumference of a circle - with long edge being diameter

Tangent and raidus at 90 degrees

Perpendicular

Visualise the two tangents at one point being same length - image other side

Same length

Three points on a circle which make a 90 degree triangle are

Three points on a circumference of a circle

If given three co ord points - how work out if on a circle?

Compare all of the lines between the points

Gradients product = -1 = circle

where the hypothesis is the diameter

Two types of reduction to linear form

Y = ax^b

Y = ab^x

Leads to difference of?

Whether the graph is plotted logy logx

Or logy x

What happens if multiply an inequality by a negative?

The sign flips!!!

Process to solve quadratic inequalities

Solve the quadratic for roots

Compare roots to shape of quadratic

Determine outcome for inequality

How to solve for k within a quadratic inequality?

Use the Discriminant!

Real roots etc will be mentioned

How to solve an exponential inequality?

ln of both sides

NEVER log of one side over to the other

X² +- 2xy + y² =

(X +- y)²

To find tangent of function

Calc gradient function

Calc x co ord

Put it into line equation to form tangent

-1 / gradient = gradient of normal

Use second differential to?

Determine type of stationary point

Second derivative <,>,=0 means?

Less than means decreasing so local maximum where going down

Greater than means local minimum as trending up

0 means potential point of inflection, but trend = 0

Potential Point of inflection

The point where the second derivative = 0

Now need to compare the gradient before and after

If gradient, first derivative, before and after = same

E.g. positive to positive or negative to negative

Then it is a point of inflection

Trend has not change, but second did reduce to 0

Concavity has changed, as second pass through 0

If before and after the potential POI gradient, first derivative, is different?

Local maxima / minima

Trend has changed to go up / down

So it has reached a max or min, stationary point, with second also at 0, but not a POI

As trend is not continued

What happens at a POI

Concavity changes

Maximum rate of change of the original graph

First derivative is at a maximum or a minimum

How to determine if concave or convex?

Pilot graph and look from below

Convex second < 0

Concave second derivative > 0

Process to follow to diagnose a point:

On wall

Diff between p(A u B) and p(A n B)

u is all overlapped so whole of both

n is intersection, where they overlap

Independent if

One doesn’t affect the other

P(A) X p(B) = P(A n B)

Mutually exclusive if

Can’t happen at same time

P(A n B) = 0

Grouped data

Data within one set which has clear distinct groups

How to ESTIMATE mean / median of groups

Midpoint of the values in the group

Modal group

Group with the largest frequency

Outliers how to classify

More than 2 SD from mean

More than 1.5 IQR from median

The WHOLE IQR, not half

Everything you need for anything binomial is in?

Formulae booklet

What does a discrete random variable sum to

sumP = 1

Conditions for a binomial distribution

All trials must be independent of each other

Probability must remain constant

There are only two outcomes

Steps in a binomial hypothesis test

Define x

Define H0

Define H1

Assume H0 is true

Define binomial expression

Test

Take critical value and compare to significance level

If significant = reject

If not significant = fail to reject

Then provide a concluding statement = there is insufficient evidence to suggest …….

Calculating the critical region

Set up the binomial distribution

Calculate the values that are WITHIN the critical region

First value to be inside it and onwards

Integration is

The reverse of differentiation

integrating and indefinite

DO NOT FORGET TO ADD C

How to integrate

Add 1 to the power

Divide that term by the new value of the power

Definite vs indefinite

Defensive to calculate area under x

Don’t need c