All core pure

quadratic formula

completed square of ax² +bx +c

a(x+b/2a)² +(c-b²/4a)

domain

set of possible inputs for a function

range

set of possible outputs of a function

coordinates of turning point

f(x)=a(x+p)² + q (-p,q)

discriminant

b²-4ac

how many roots does b²-4ac> 0

2 distinct real roots

how many roots does b²-4ac= 0

repeated solution,tangent

how many roots does b²-4ac< 0

no real roots,no touching of x axis

z/

integers

E

belongs to

N

natural numbers-positive integers

r

any number-real

positive cubic graph

negative cubic graph

positive quartic graph

w shae

negative quartic graph

where are the asymptkotes if graph y=k/x and k/x²

x=0 y=0

y=k/x

y=k/x²

asymptote

line which the graph approaches but never reaches

what if repeated root

tangent

y=f(x) + a

move vertically up

y+f(x+a)

move horizontally left

y=af(x)

times y by a vertical

y=f(ax)

timed x by 1/a horizontal

y=-f(x)

reflection in x

y=f(-x)

reflection in y

equation of circle centre (a,b) and radius r

(x-a)²+(y-b)²=r²

equation of circle in other form

x²+y²+2fx+2gy+c=0

centre (-f,-g)

radius √f²+g²-c

complete square

if prq angle is 90

then r lies on circle with diameter pq

what is angle in semicircle

right angle

cosine rule to find missing side

a²=b²+c²-2bc cos A

cosine rule to find angle

cos A= b²+c²-a² /2bc

sine rule to find angle

a/sinA = b/sinB = c/sinC

sine rule to find missing angle

sin A/a = sin B/b =sinC/c

2 angles from sine rule

can be both obtuse and acute so to work out other sinx=sin(180-x)

area of triangle

1/2ab sinC

sin graph

• repeats every 360 and crosses the x axis at …-180,0,180,360

• has a maximum value of 1 and minimum value of -1

cos graph

• repeats eveyr 360 and crosses x axis at …-80,90,270,450…

• has maximum value of 1 and minimum value of -1

tan graph

• repeats every 180 and crosses x axis at …-180,0,180,360…

• has no maximum or minimum values

• has vertical asymptotes at x=-90,x=90,x=270…

angle quadrants

• CAST

• all positive in fist quadrant

• only sin positive in 2nd

• only tan positive in third

• only cos positive in 4th

angle size in firts quadrant

θ

angle size in 2nd quadrant

• 180 - θ

• sin(180-θ)=sin θ

• cos(180-θ)=-cosθ

• tan(180-θ)=-tan θ

angle size in 3rd quadrant

• 180 + θ

• sin(180+θ)=-sin θ

• cos(180+θ)=-cosθ

• tan(180+θ)=tan θ

angle size in 4th quadrant

• 360 - θ

• sin(360-θ)=-sin θ

• cos(360-θ)=cosθ

• tan(360-θ)=-tan θ

sin² and cos² identity

sin²θ+cos²θ= 1

tan,cos,sin identity

tanθ=sinθ/cosθ

when do silutions only exist when sinθ=k and cosθ=k

-1<k<1

what values exist for tanθ=p

all values of p

principal value

when you use inverse trigonometric function

range of principal value for cos^-1

0<θ<180

range of principal values for sin^-1

-90<θ<90

rang eof principal values of tan^-1

-90<θ<90

unit vector

vector of length 1

unit vector in direction of a

a/│a│

^

a

unit vector

differentiating from first principles

gradient of tangent to curve

same as gradient at point

gradient of normal at point

negative reciprocal of the curve at that point

increasing function

f’(x)>0

decreasing function

f’(x)<0

gradient of stationary point

0

max point

d²y/dx²<0

min point

d²y/dx²>0

point of infliction

d²y/dx² = o then checj points either side

how to skect max/min point on gradietn function grah

points cross x axis

how to skect point of inflection on gradietn function grah

touch x axis

how to skect positive gradient on gradietn function grah

above x axis

how to skectnegative gradient on gradietn function grah

below x axis

how to skect vertical assymptote on gradietn function grah

vertical assymptote

how to skect horizontal assymptote on gradietn function grah

assymptote at y =o x axis

Cylinder surface area

2PIrh +2Pi r²

what must be added when integrated

+c

y=a^x graph

y=-a^x

y=1/a ^x

y=-1/a ^x

a^x=n in logs

what does log a base a equals

1

what does log 1 base a equals

0

graph of y=ln x

reflection of graph y=e^x in line y=x

what is e^lnx equivalent to

ln(e^x)=x

lnx in log

log x base e

ln(x)=b as e

e^b =x

ln(-x) graph

-ln(x) graph

how to draw modulus function

y=|ax+b| sketch y=ax + b then reflect section below x axis in x axis

sketch the graph of y = f(|x|)

Sketch the graph of y = f(x) for x > 0. • Reflect this in the y-axis.

standard deviation

square root of variance σ

variance

σ²

standard deviation equation

√∑(x-x̄)²/n = √∑x²/n -(∑x/n)² =√sxx/n

mean of coded data

ȳ=x̄-a/b

standard deviation of coded data

σy=σx/b where σx is standard deviation of original data

How to write X as a normally distributed random variable

X ~ N(µ,σ²)

inverse normal distribution

P(X<a)=p

standard notmal distribution

• mean 0 and standard deviation 1

• Z~N(0,1²)

formula to code variable x that is ormally distributed into standard normal distribution

z=x-μ / σ

how can the binomial distribution be approximated

• μ=np

• σ=√np(1-p)

percentage error equation

approx-exact/exact x 100