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
gradient equation
y₂-y₁/x₂-x₁
parallel
same gradient
perpendicular
negative reciprocals m₂ = -1/m₁
Point-gradient form
y-y₁ = m(x-x₁)
gradient-intercept form
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'
radical
An expression made up of a radical sign and a radicand
surd
a real, irrational radical
√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
discriminant of a quadratic
∆ = 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
Addition Law of Probability
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
positive definite quadratics
- quadratics that are positive for all values of x- a>0 and ∆<0
negative definitie quadratics
- 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?
Completing the square (quadratics)
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→radians
degrees ×π/180
radians→degrees
radians ×180/π
arc length in radians
l=θr
Area in radians
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)