IB Math SL Key Words

arithmetic sequence

a sequence of numbers in which each term differs from the previous one by the same fixed number

How to find a term in an arithmetic sequence

uⁿ = u₁ + (n-1)d

where d is the common difference

algebraic definition of arithmetic sequence

uⁿ⁺¹ - uⁿ = d

where d is an integer known as the common difference

Geometric sequence

a sequence of numbers where each term can be obtained from the previous one by multiplying by the same non zero constant

algebraic definition of geometric sequence

(uⁿ⁺¹) ÷ uⁿ = r

where r is a constant called the common ratio

how to find a term in a geometric sequence

uⁿ = u¹× r^(n-1)

where r is a constant called the common ratio

series

the addition of the terms of a sequence

sum of a series

the result of the addition of the terms of a sequence

sigma notation

sum of an an arithmetic series

Sⁿ=

(n÷2)(u¹+uⁿ)

or

(n÷2)(2u¹+(n-1)d)

sum of a geometric series

Sⁿ=

(u₁(rⁿ-1))÷(r-1)

or

(u₁(1-rⁿ))÷(1-r)

domain of a function

the set of permissible values that x may have

range of a function

the set of permissible values that y may have

composite function

one function acting on the result of another function

f(g(x)) or (f ₀ g)(x)

how to make a sign diagram

a horizontal line (x axis)

critical values (x intercepts or vertical asymtotes)

positive or negative signs

what happens when you raise a negative number to an odd power?

the result is negaitve

what happens when you raise a negative number to an even power?

the result is positive

a^(m÷n) =

ⁿ√a^m

log(c)A + log(c)B =

log(c)(AB)

log(c)A - log(c)B =

log(c)(A÷B)

n×log(c)A =

log(c)(Aⁿ)

change of base rule

log(b)A = log(c)A ÷ log(c)b

linear equation

f(x) = ax+b

quadratic equation

f(x) = ax² + bx + c

cubic equation

f(x) = ax³ + bx² + cx + d

exponential equation

f(x) = a^x

what are the two translations

y=f(x) + b

and

y=f(x-a)

what does y=f(x) + b do?

translates the graph vertically, b

what does y=f(x-a) do?

translates the graph horizontally, a

what are the two stretches?

y=pf(x)

and

y=f(x/q)

what does y=pf(x) do?

it stretches the line vertically, p>1 then points move further away from x axis

what does y=f(x/q) do?

it stretches the line horizontally, q>1 then the points move further away from y axis

what are the two reflections

y=-f(x)

and

y=f(-x)

what does y=-f(x) do?

reflect across the x axis

what does y=f(-x) do?

reflect across the y axis

what is the discriminant?

b² - 4ac

also known as ∆

what does it mean if ∆=0

then x=-b÷(2a) is the only solution

and the graph touches the x axis

what does it mean if ∆>0

then there are two distinct solutions

and the graph cuts the x-axis twice

what does it mean if ∆

there are no real roots

and the graph does not cut the x-axis

talk about

y = a(x-p)(x-q)

form

x-intercepts are p and q and vertex is

(p+q)÷2

and

f((p+q)÷2)

talk about

y = a(x-h)²

form

touches x axis at h, vertex is (h,0)

talk about

y = a(x-h)² + k

form

vertex is (h,k)

talk about

y = ax² + bx + c

form

y-intercept is c

axis of symmetry is (-b÷2a)

how to find the minimum value of a quadratic equation

if a>0 the minimum value of y is at x=-(b÷2a)

how to find the maximum value of a quadratic equation

if a

what is the period of a function

the length of one repitition

how to find the amplitude of a function

(max-min)÷2

how to find the principal axis of a function

y = (max+min)÷2

what is the principal axis of a function

the line in the middle of a wave

what is the amplitude of a function

the distance between a maximum or minimum point and the principal axis

general functions of the sine graph

y = a×sin (b(x-c)) + d

a will affect: amplitude

b will affect: period (2π/b)

c will affect: horizontal translation

d will affect: vertical translation

what is cos(x) in terms of sin(x)

cos(x) = sin(x+(π/2))

how to find the period of a sin or cos equation

2π/b

how to find the period of a tan equation

π/b

what is a scalar

quantity that only has magnitude

what is a vector

a quantity that has both magnitude and direction

state that the magnitude of a vector is its length

he magnitude of a vector is its length

how to find the algebraic length of a vector

|v| = √v₁² + v₂²

how to find the distance between points A and B

AB = √(x₂-x₁)² + (y₂-y₁)²

dot product of two vectors

v dot w = v₁w₁ + v₂w₂ + v₃w₃

how to find the angle between vectors

cosθ = v dot w divided by |v|×|w|

vector equation of a line

r = a + t×b, t∈R

how to find the cartesian equation of a vector line

equate t to

(x-a₁)/b₁

and

(y-a₂)/b₂

how to find the parametric equation of a vector line

x = a₁ + tb₁

y = a₂ + tb₂

z = a₃ + tb³

what the vector equation means in terms of real world problems

r = a + tb

if a body has initial position vector "a" and moves with constant velocity "b" then its position at time "t" is given by that equation

what happens if you dot product perpendicular vectors?

You get zero which means this is useful in order to find two perpendicular vectors

what is the lower quartile

the bottom 25% of values, Q₁

what is the upper quartile

the top 25% of values, Q₃

how to find the IQR

IQR = Q₃ - Q₁

number of trials definition

total number of times the experiment is repeated

outcome definition

the different possible results for one trial

frequency definition

the number of times this outcome has been observed

relative frequency definition

frequency of an outcome expressed as a fraction or percentage of total outcomes

complementary events

two events are complementary if their probabilities add up to one

independent events

two events are independent if the occurrence of each of them does not affect the probability of the other

if A and B are independent events then

P(A and B) = ?

P(A) × P(B)

dependent events

two or more events are dependent if they are not independent

If A and B are dependent events then P(A and B) = ?

P(A) × P(B given that A has already occurred)

what does P(A∪B) mean?

probability of A or B or both happening

what does P(A∩B) mean?

probability of where A and B meet happening

what does P(A|B) mean?

probability of A given B. If B happens for sure what is the probability of A.

for two events A and B;

P(A∪B) = ?

P(A) + P(B) - P(A∩B)

if A and B are mutually exclusive then;

P(A∩B) = ?

0

for two events A and B;

P(A|B) = ?

P(A∩B) ÷ P(B)

for two events A and B;

P(A∩B) = ?

P(A|B) × P(B) or P(B|A) × P(A)

if A and B are mutually exclusive then

P(A|B) =

P(A)

differentiation from first principles

f '(x) = (f(x+h) - f(x)) ÷ h

all as h approaches 0

∫kƒ(x)dx

can be changed to

k∫ƒ(x)dx

For the sum of integrals

∫ƒ(x) +g(x)dx = ?

∫ƒ(x)dx + ∫g(x)dx

If a and b are constants then,

∫ƒ(ax+b)dx = ?

(1/a)∫ƒ(ax+b)

what does distance mean in terms of integrals?

how much ground something has covered

what does displacement mean in terms of integrals

how far something has moved from the origin

what does velocity mean in terms of integrals

the rate of change of displacement with respect to time

how do you remember the order of integration and differentiation for displacement velocity and acceleration?

Dear Daddy, Vests Are Inteligent

Differentiation, Displacement, Velocity, Acceleration, Integration

unit circle