From patterns to generalizations
Sequences and series
Proof
Arithmetic
u + (n-1)d
Common difference = d
Finite or infinite
Sum formula is in the data booklet. u of n can be written as (n-1)d
Geometric
Common ration = r
u*r^(n-1)
Finite sum has one formula
Infinite sum
A logical set of steps that validates the truth of a general statement beyond any doubt.
Direct proof
Identify the statement
Use axioms, theorems to make deductions that prove the conclusion of the statement to hold
Proof by contradiction
QED; RHS = LHS
Identify what is being implied by the statement
Assume the implication is false
Use axiom and theorems to arrive at the contradiction
Proves the original statement must be true
E.g. root 2
Induction
Basic step
Inductive step
Counting principles and the binomial theorem
Assumption
Factorial
Permutations
Combinations
Binomial expansion
The binomial expression is generalized for: negative numbers and fractions. Refer to pages 62 to 63 of the Oxford Book.
Counterexample
Provide a numerical example that makes the statement false
Complex numbers
Quadratic equation and inequalities
Solve by factorization
Solve by completing the square
Quadratic function
Discriminant
more than 0: two answers
= 0: one answer
smaller than 0: no real answers
Complex numbers
a+bi
Geometric approach
Real part Re(Z) = a
Imaginary part Re(Z) = b
x axis is Re
y axis is Im
Modulus (absolute value)
Distance from the origin (considering geometric approach)
Two complex numbers are equal if their real parts are equal AND their imaginary parts are equal
Conjugate
Adding a conjugate to its complex number gives a
Subtracting a conjugate from its complex number gives b
How to find: a=a, b=-b
i^2=-1; i^3=-i; i^4=1; i^5=i
To find the root of Z, square Z and (a+bi) and then find a and b
Polynomial equations and inequalities
Polynomials are equal only if they have the same degree and coefficients
Linear combination of n functions
Factor theorem
For any two polynomials f and g there is:
Order does not matter
Order matters
Find the term independent of x
Representing relationships: functions
A relation is a function if...
It acts on all elements of the domain
It is well-defined, meaning it pairs every element of the domain with one and only one y-value (range)
Special functions
Quadratic
Standard form
Vertex form (h,k):
Intercept form:
Find the axis of symmetry (which is also the x value at the vertex):
Rational
y=(1/x) is an example
Radical
Absolute value
Partial fractions
Factorize the denominator, find the values of A and B (substituting values that makes part of the expression 0)
Piecewise-defined function
Example
Classification of functions
General groups
One-to-one: every element in the range corresponds to exactly one element in the domain
Many to one: more than one element in the domain has more than one image
Onto: similar to many to one, but needs to be defined at every value of its set range
Even and odd
Even: f(x) = f(-x); symmetrical about the x-axis
Odd: f(-x) = -f(x); rotational symmetry of 180 degrees about the origin
Composition of functions
g(f(x)) or gof
Identity function: function f that when composed with g leaves g unchanged: f(x) = x
Inverse function
Find it by interchanging the y and x
Function h that, when composed with g, yields the identity function
Self-inverse
Operation with functions
For y=f(x) to y=|f(x)|, y>0 remains and y<0 is reflected about the x-axis
For y=f(x) to y=f(|x|), x>0 remains and x<0 replaced by a reflection of x<0 in the x-axis
Reciprocal
Zeros turn into vertical asymptotes
The y-intercept b turns into 1/b
Minumum value of y turns into maximum
Greater and smaller than zero at the same x-values as the original
Approaching 0 turns into approaching infinity
Transformation
a is the vertical dilation (stretched by a factor of b)
b is the horizontal dilation (stretched by a factor of 1/b)
c is the horizontal translation (negative value moves right)
d is the vertical translation (positive value moves up)
The rational function can be transformed
Zeros
If a polynomial has a complex zero Z, then the conjugate is also a zero of the function
Addition of zeros
Multiplication of zeros
Simultaneous equations
Two linear equations
Unique pair (lines intersect)
No real numbers (lines are parallel)
Infinitely many pairs (the lines coincide)
Three linear equations (if you have three points of a quadratic function, turn them into equations and solve like this!)
Unique triplet
Not triplet
Infinitely many triplets
IBDP SEMESTER 1 MATHEMATICS AA HL