**Core 1**

**Curves & Translations**

Quadratics

**Co-ordinate Geometry**

**by Luke Storry 2011**

https://portal.exe-coll.ac.uk/courses/mas/maths/asm/01d/Specification/MEI%20Core%20Maths%201%20Specification.pdf

For 13th Jan 2012 Exam

Specification:

**Revision Notes**

**Polynomials**

**Uncertainty**

**Indices**

d > 0 -> 2 distinct roots

d = 0 -> 2 equal roots (aka just 1)

d < 0 -> 0 real roots

Gradients

Distances

A

B

Dx

Dy

using pythagoras,

Straight Lines

Equation

perpendicular gradients have a product of -1, so

Circles

if (a,b) is the origin and (x,y) is a point on the circle, then:

To find origin from three points, find equation of two lines through those points, then the perp.bisectors will intersect on the origin

Parallel lines have EQUAL gradients; only 'C' changes

to find points of intersection (or if there are any) just do a simulatanious equation with them, to see if any values meet both.

Order of Polynomials

is the highest power of x in it.

eg quadrilaterals are 2nd order, and cubics are 3rd

let

be any polynomial

function of x

F(x)

Factor Theorem

f(a) =0 <=> (x-a) is a factor of f(x)

f(a/b) = 0 <=> (bx-a) is a factor of f(x)

Remainder Theorem

when

f(x)

is divided by

(x-a)

, the remainder is

f(a)

Solving

& sketching the graph

1) factorise (divide) down to roots

2) draw curve

3) draw x axis through it at appropriate place

4) mark on roots

5) draw y axis through ti at appropriate place

Operations

+&-

x

just do the coefficients

Box method

/

reverse box plot

Binomial Expansions

nCr w/o calc

- NEED TO KNOW!!

where

Quadratic

has vertex

line of symmetry

y intercept

( -p , q )

x = -p

( 0 , c )

f(x) = ax^3 + bx^2 + cx + d

Cubic

Completed square

A

translation

of a function may be given by:

Translations

f(x - t) + s

this is given by the vector

t

s

()

horizontal (x)

vertical (y)

can be solved like linear equations,

Linear Inequalities

Solve by drawing it!

Quadratic Inequalities

except that when multiplied or divided by a negative, the sign reverses

then identify the part(s) of the x axis that complies with the equation.

if the denominator of a fraction is sqrt{b},

multiply top and bottom by sqrt{b}

Rationalising Denominators

if the denominator is (a + sqrt{b}), then

multiply top and bottom by (a - sqrt{b})

look for square factors

Simplifying square roots

Positive "a" (x coefficient)

means a smily face,

and visa versa

Smily / Frowny

vertex (aka turning point) is also at

x= -b/2a

**Logic**

-1/grad = perp. grad

Basically, if no remainder, it must be a factor

eg. 7C4

= 7! /(4! 3!)

=> implies

<= is implied by

<=> both^ (equivilant to)