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Maths Core 1

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Daniela Schenk

on 3 January 2012

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Transcript of Maths Core 1

Maths core 1
Basic
Algebra
Manipulating
Algebraic
Expressions
Collecting like terms
Removing /expanding brackets
Factorising with one bracket
Multiplication
Algebraic fractions
Linear Equations and
Rearranging Formulae
Solve linear equations using
a variety of methods
Rearranging formula to
change the subject
e.g: make c the subject of:
Quadratic
Equations
3 different types of quadratic equations:
1.
2.
3.
3 ways to
solve

1. Factorising:
2. Completing the square:
e.g:
3. Quadratic Formula
e.g:
Using the
discriminant
(d)
and
Simultaneous
Equations
Solve by elimination
Solve by substitution
1. Make coefficient of either y or x equal in both equations by multiplying the equation by the relevant number.
2. Subtract equation 2 from equation 1 to find the value of either y or x
3. Substitute the value into either equation 1 or equation 2 to find the value of the other variable.
1. If y isn’t already the subject, make y the subject of the equation
2. Substitute y
3. Solve the quadratic equation using preferred method.

Co-ordinate
Geometry
Equations of
Straight Lines
Passing through (0,c)
Passing through (x1,y1)
Straight line with equation:
ax+by=c
Passes through (c/a,0) and (0, c/b)
Gradients of
Straight Lines
Using point (a,b)
Gradient of Straight Line:
Need to know 2 points on the line
e.g for
Parallel lines have equal graideints
e.g:
y=2x+3
y=2x-1
Perpendicular lines:
Gradients multiply to give -1
m
x
m
= -1
Positive and negative gradients
Positive gradient
-e.g y=2x
Negative gradient
-e.g y=-2x
Distances and
Mid-points
-Pythagoras theorem
-Average
Circles and
other Curves
Equations of circles:
With centre (0,0) and radius r
With centre (a,b) and radius r
Polynomials
Curves and
Translations
Uncertainties
Indices
Operations with
Polynomials
Polynomial
functions
Factor and
Remainder
Theorems
Solving equations/
Graph sketching
Binomial Expansion
Quadratic
functions
Cubic
functions
Translations
Working with
Powers
and roots

Laws of
Indices

Linear
Inequalities

Points of intersection:
Finding the centre and radius from a sketch:
Adding Polynomials
Subtracting Polynomials
Factor theorem
Remainder theorem
Quadratic
Inequalities
Rationalising the denominator
Simplifying square roots
e.g
e.g
e.g
e.g
e.g
e.g
Solve by using simultaneous equation, substitute equation of other line or curve into equation of circle
e.g:
Centre: (
3
,
-4
) Radius:
6
Centre: midpoint of diameter
Radius: distance between centre and any other point on circle
Add together the terms
e.g
Subtract the terms
e.g
Multiplying Polynomials
Dividing Polynomials
Formulae
Example
hint
Always write it down in terms of a and b before substituting in values
Pascal's Triangle
(how you can find nCr)
Order/degree
Polynomials
As functions of x
Polynomial:
The power of x is an integer that is greater than or equal to 0
Distances:
A
B
Dx
Dy
(a)
(b)
Midpoint:
(x1,y1)
(x2,y2)
(a,b)=
A polynomial may be divided by linear functions. To divide a polynomial rearrange the equation so you are multiplying by the answer:
Linear inequalities can be handeled just like linear equations, but when both sides are multiplied or divided by a negative number then the inequality must be reversed.
these can be solved by solving the corresponding quadratic equation, DRAWING the graph and then finding the solution set(s)
3 different forms that the
equations can be in
1.
2.
3.
1. Finding the y intercept:
Finding points on a quadratic curve:
For:
e.g:
Graph:
Make x=0 and solve
So y intercept: (0,7)
2. Finding the root(s):
Make y=0
Expand brackets
Find a,b,c
Find d
Use the quadrtatic formula
to find the roots
So x intercepts: (-1,0) and (-7,0)
d<0 so 2 distinct roots
3. Finding the vertex (turning point):
So vertex: (-4,-9)
Line of symmetry: x=-4
Finding points on a quadratic curve:
For:
e.g:
Graph:
1. Finding the y intercept:
The value of c is the y intercept
+0
In this case y intercept: (0,0)
2. Finding the root(s):
Make y=0
Find a,b,c
Find d
Use the quadrtatic formula
to find the roots
So x intercepts: (0,0) and (-4,0)
3. Finding the vertex (turning point):
So vertex: (-2,-4)
Line of symmetry: x=-2
Finding points on a quadratic curve:
For:
e.g:
Graph:
1. Finding the y intercept:
Make x=0 and solve
So y intercept: (0,-4)
Look at the brackets,
the values that make them =0
are the x intercepts
So x intercepts: (-4,0) and (-1,0)
3. Finding the vertex (turning point):
So vertex: (-1.5,-6.25)
2. Finding the root(s):
Basic Parabolas:
When multiplying polynomials, draw out a grid and put in the polynomials, then multiply each term...
e.g:
x
...then add all the terms with the same order to getthe answer
So:
e.g:
So
By working backwards throught the table and sing multiplication you can work out the answer making sure that the answer to the multiplication is the same as the polynomial that is being divided
So:
Example of a cubic curve
Can be factorised using factor theorem to find one factor, then divide original equation by found factor to the the factor and a quadratic function, then factorise again to get the cubic function factorised. Then a graph can be sketched using that information. (see solvin equations/sketching graphs)
a<0
a>0
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