# Math AA SL

N

the set of positive integers and zero

Z

the set of integers

Z⁺

set of all positive integers Z⁺

Q

the set of rational numbers

Q'

the set of irrational numbers

R

the set of real numbers

Square Perimeter

P = 4l

Rectangle Perimeter

P = 2(l+w)

Circle Circumference

C = 2πrC = πd

Arc Length

L = rθL = (θ/360)2πr

Circle Area

A = πr²

Sector Area

A = (θ/360) x πr²

y₂-y₁/x₂-x₁

parallel

perpendicular

negative reciprocals m₂ = -1/m₁

y-y₁ = m(x-x₁)

y = mx + b

general form

ax + by = d

midpoint formula

M = (x₁+x₂)/2, (y₁+y₂)/2

Solution by Substitution (straight lines)

substituting (X or Y) in an expressionfind x = (...) and put x = (...) into y+x=(..)

Solution by elimination (straight lines)

make the coefficients of x or y the same size but opposite sign, then add the equations

set

a collection of numbers or objects

element

object of a set

is an element of

is not an element of

n(A)

the number of elements in set A

finite set

a set with a given number of elements

infinite set

a set with infinite number of elements

subset

Every element of A is also an element of B. A ⊆ B

proper subset

if every element of A is also an element of B, but A ≠B. A ⊂ B

empty set

a set with no elements, ∅

intersection of sets

set of elements in both A and B. A ∩ B

disjoint (mutually exclusive)

sets have no elements in common

union of sets

set of elements in either A or BA ∪ B

complement of a set

all of the elements NOT in that set A and A'

surd

√ab = √a ×√b

√ab = √a ×√b

√a/b = √a/√b

√a/b = √a/√b

Rationalise the denominator

b/√a → √a/√a

-2¹

a negative base raised to an odd power is negative

-2²

a negative base raised to an even power is postive

product rule (exponents)

a° × aⁿ = a°⁺ⁿ

Quotient Rule (Exponents)

a°/aⁿ = a°⁻ⁿ

power rule (exponents)

(a°)ⁿ = a°ⁿ

Product to a power (exponents)

(ab)ⁿ = aⁿbⁿ

product to a quotient (exponents)

(a/b)ⁿ = aⁿ/bⁿ

zero rule (exponents)

a⁰ = 1, a ≠ 0

negative power (exponents)

a⁻ⁿ = 1/aⁿ and 1/a⁻ⁿ = aⁿ

scientific notation

a × 10ⁿ

power equation

xⁿ = k, n≠0

∆ = b-4ac

number sequence

an ordered list of numbers defined by a rule

General term of a sequence

uₙ - represents the sequence that can be generated by using uₙ as the nth term

arithmetic sequence

An arithmetic sequence is created by adding or subtracting a common difference.arithmetic sequence↔uₙ+1-uₙ=d

general term formula (arithmetic sequence)

uₙ = u₁+(n-1)d

geometric sequence

A geometric sequence is created by multiplying (or dividing) a common ratio. geometric sequence↔uₙ+1/uₙ = r

general term formula (geometric sequence)

uₙ = u₁rⁿ⁻¹

growth and decay (sequences)

uₙ = u₀ ×rⁿ

series

Sum of terms in sequence

sigma notation

for any sequence a₁, a₂, a,..., the sum of the first k terms may be written k∑ aₙn=1which is read "the sum of all numbers form Uₙ where n = 1,2,3, ..., up to k.

Properties of sigma notation

0

arithmetic series

the sum of the terms of an arithmetic sequenceSₙ=n/2(u₁+uₙ) orSₙ = n/2(2u₁+(n-1)d)

finite geometric series

the sum of terms in geometric sequenceSₙ = u₁(rⁿ-1)/r-1 or Sₙ = u₁(1-rⁿ)/1=r

infinite geometric series

S = u₁/1-r

supplementary angles

two angles whose measures equal 180 degrees

Area of a non-right triangle

A = ½absinC

Cosine rule

a²=b²+c²-2bc cosA or cosA = b²+c² - a² / 2bc

Sine rule

a/sinA = b/sinB = c/sinC or SinA/a = sinB/b = sinC/c

experimental probability =

#NAME?

Theoretical Probability

P(event)=(Number of favorable outcomes)/(Total possible outcomes) andP(A) + P(A') = 1

P(A∪B) = P(A)+P(B)-P(A∩B)disjoint , P(A∪B) = P(A)+P(B)

independent events

P(A∩B)=P(A)×P(B)

dependent events

events in which the occurrence of one event affects the probability of the otherP(A∩B) = P(A)×P(B given A has occured)

Conditional Probability

P(A|B)=P(A∩B)/P(B)

formal definition of independence

A and B are independent events ↔ P(A∩B)=P(A)P(B)

sampling error

when characteristics of sample differs from that of the whole population

measurement error

inaccuracies in measurement at the data collection stage or a loaded question

coverage error

small or biased samples

non-response errors

when a large number of people selected for a survey choose not to respond

simple random sampling

every member of the population has equal probability to be in sample

systematic sampling

sample is created by selecting members of the population at regular intervals

convenience sampling

members are chosen simply because they are easier to select or more likely to respond

stratified sampling

when the population can be divided into subgroups - randomly selected

quote sampling

when the population can be divided into subgroups - specifically selected

categorical variable

describes a particular quality or characteristic

quantitive variable

variable has a numerical value

discrete variable

takes exact number values

continuous variable

can take any numerical value within a certain range

mode

Data value that occurs most often in a data set

mean

statistical name for averagek∑xₙn=1-----n

median

middle value of an ordered setn+1/2

range (statistics)

range = maximum - minimum

interquartile range

interquartile range = upper quartile - lower quartileIQR = Q₃ - Q₁

box and whisker diagrams

...

outliers (upper)

upper boundary = upper quartile + 1.5 × IQR

outliers (lower)

lower boundary = lower quartile - 1.5 × IQR

Variance

measures the average degree to which each number is different from the meanσ² = k ∑(xₙ-µ)² n=1 ------- k

standard deviation

looks at how far from the mean a group of numbers is, by using the square root of the varianceσ=√k ∑(xₙ-µ)² n=1 -------- k

- quadratics that are positive for all values of x- a>0 and ∆<0

- quadratics which are negative for all values of x- a<0 and ∆<0

sign diagram

indicates the values of x for which a function is negative, zero, positive, undefined- horizontal line that represents x-axis- (+) and (-) signs indicating where the graph is above and below axis- the zeros of the function

function

a relation in which no two different ordered pairs have the same x-coordinate

function notation

using "f(x)=" to represent "y=

domain (functions)

set of values that the variable on the horizontal axis can take

range (functions)

set of values that the variable on the vertical axis can take

Reciprocal Function

y = k/x, k≠0

y = b/cx+d + a

VA is x=-d/cHA is y=a

y = ax+b/cx+d

VA is x=-d/cHA is y = a/c

composite function

a combination of two functions such that the output from the first function becomes the input for the second function. (a function in a function)

inverse function

The function that results from exchanging the domain (x-values) and range (y-values) of a one-to-one function.

Translations of Functions

y=f(x) + b (vertical)y=f(x-a) (horizontal)

stretches of functions

y=pf(x) (vertical)y=f(qx) (horizontal) stretch factor being 1/q

Reflections of Functions

y=-f(x) (x-axis)y=f(-x) (y-axis)

exponential equation

An equation where the unknown is in the variable

exponential functions

y = a^x family of functions

y=p×a^x-h+k

- a controls steepness- h controls horizontal translations- k controls vertical translations- HA is y=k

logs in base n

n^x=b ↔ x=logₙb

log m + log n

#NAME?

y=x²-6x+7y=x²-6x+3²+7-3²y= (x-3)² - 2

log m - log n

0

nlogb

#NAME?

ln(e^x)

e^lnx = x

change base rule logs

log₀a=logₙa/logₙb

degrees ×π/180

l=θr

a=½θr²

pythagrean identity

cos²θ + sin²θ = 1

Sine Function Equation

y=asin(b(x-c)) + d

Cosine Function Equation

y=acos(b(x-c)) +d

tangent function equation

y=atan(b(x-c)) +d

negative angle formulae

sin(-θ) = -sinθcos(-θ) = cosθ

supplementary angle formulae

sin(π-θ) = sinθcos(π-θ) = -cosθ

complementary angle formulae

sin(π/2- θ) = cosθcos(π/2 - θ) = sinθ

cos²θ + sin²θ = 1

sin²θ = 1 - cos²θcos²θ = 1 - sin²θ

Double Angle Identities

sin(2θ)=2sinθcosθcos(2θ)=cos²θ-sin²θcos(2θ)=2cos²θ-1cos(2θ)=1-2sin²θtan(2θ)=(2tanθ)/(1-tan²θ)