### Topic 1: Algebra

#### Key concepts to learn:

##### SL and HL:

- Arithmetic and geometric sequences and series
- Sigma notation (Σ)

- Working with exponents and logarithms
- Change of base formula:

- The binomial theorem, permutations and combinatorics
- Pascal's triangle

##### HL only:

- Proof by mathematical induction
- Complex numbers in cartesian form
- Complex numbers in modulus-argument form
- Euler's formula:

- Raising complex numbers to arbitrary powers
- Conjugate roots of polynomials
- Solving systems of linear equations with up to three unknowns

#### Helpful learning resources:

Check out the rest of Khan Academy's sequences series, and induction playlist here.

Check out the rest of Khan Academy's polynomials playlist here.

Check out the rest of Khan Academy's logarithm playlist here.

Check out the rest of Khan Academy's complex numbers playlist here.

### Topic 2: Functions and equations

#### Key concepts to learn:

##### SL and HL:

- Domain and range of a function
- Composite functions
- Inverse functions
- Transformation of graphs; shifting, stretching and reflecting function curves
- Max. and min. value of functions, intercepts and asymptotes
- The quadratic function
- Using the quadratic formula:
- The discriminant determines if function has one, two or zero real roots:

- Reciprocal and rational functions
- Graphs of exponential and logarithmic functions
- Understanding the relationship:

##### HL only:

- The fundamental theorem of algebra
- Any polynomial of nth degree terms has exactly n roots (which may be real or complex)

- The factor and remainder theorems for solving 3rd and 4th degree equations
- The remainder when you divide a function by a number is the same as the value of the function evaluated at that number

- The sum and product of roots of polynomial equations

#### Helpful learning resources:

Check out the rest of Khan Academy's advanced functions playlist here.

Check out the rest of Khan Academy's polynomials playlist here.

### Topic 3: Circular functions and trigonometry

#### Key concepts to learn:

##### SL and HL:

- The unit circle; measuring angles in radians
- radians
- Finding length of an arc and area of a sector from central angle of a circle in radians

- Understanding and in terms of the unit circle
- Knowing exact trig. values of common angles
- The Pythagorean identity
- Double angle identities for sine and cosine
- Using , and as functions; understanding their domain, range and amplitude.
- Manipulating functions of the form

- Solving quadratic equations involving trig. ratios
- Solving for triangles using trig. ratios
- The cosine rule:
- The sine rule:
- Area of a triangle:

##### HL only:

- The reciprocal trig. ratios , and
- Extra Pythagorean identities
- The inverse trig. functions , and

#### Helpful learning resources:

Check out the rest of Khan Academy's unit circle playlist here.

Check out the rest of Khan Academy's graphs of trig functions playlist here.

Check out the rest of Khan Academy's trig identities playlist here.

### Topic 4: Vectors

#### Key concepts to learn:

##### SL and HL:

- What a vector is
- Representing a vector as

- Algebraic and geometric manipulation of vectors
- Sum and difference of vectors
- Multiplying a vector by a scalar
- Magnitude of a vector:

- The scalar product (dot product) of two vectors
- Vectors that are parallel or perpendicular to each other
- The angle between two vectors
- Vector equations of the form
- Finding point of intersection between two points

##### HL only:

- The vector product (cross product) of two vectors
- Vector equation of a plane:
- Cartesian equation of a plane:
- Intersections and angles between one, two or three planes

#### Helpful learning resources:

Check out the rest of Khan Academy's basic vectors playlist here.

Check out the rest of Khan Academy's advanced vectors playlist here.

### Topic 5: Statistics and probability

#### Key concepts to learn:

##### SL and HL:

- Difference between a population and a sample
- Difference between discrete and continuous data
- Representing data using frequency distribution tables and histograms
- Mean, median and mode
- Mean: the average value of the data
- Median: the middle value of the data
- Mode: the most commonly occuring value in the data

- Range, interquartile range, variance and standard deviation
- Cumulative frequency graphs
- Understanding trials, outcomes, events and equally likely outcomes
- U (the sample space) is the total number of outcomes that can possibly happen
- The probability of an event A is

- Complementary events: A and A' ("not A")
- Using Venn and tree diagrams to find probabilities visually
- Combined events,
- Mutually exclusive events in which
- Conditional probability
- Independent events
- In which '

- Discrete and random variables
- Expected value E(X) for discrete data
- Binomial distribution for finding probability
- Normal distributions

##### HL only:

- Bayes' theorem for up to three events
- Poisson distribution

#### Helpful learning resources:

Check out the rest of Khan Academy's descriptive statistics playlist here.

Check out the rest of Khan Academy's data and statistics playlist here.

Check out the rest of Khan Academy's dependent and independent probability playlist here.

Check out the rest of Khan Academy's random variables playlist here.

### Topic 6: Calculus

#### Key concepts to learn:

##### SL and HL:

- Understanding limits
- Understanding definition of derivative from limits
- '

- Understanding the derivative as the slope of the tangent line and as the rate of change of a function.
- Equations of tangent and normal lines
- The slope of a normal line to a point on a curve is the negative reciprocal of that point's tangent line

- The chain rule for taking derivatives of functions within functions
- The product and quotient rules
- The product rule:
- The quotient rule:

- The second and third derivative
- Finding min. and max. points of a function, as well as points of inflexion
- When a function's derivative is equal to zero at a point, that point is a min. point, max. point or a point of inflexion.

- Finding indefinite integrals (or anti-derivatives)
- For example,

- Definite integrals
- The fundamental theorem of calculus: '

- Definite integrals to find area under the curve
- Finding volumes of revolution when a graph is rotated around the x- or y-axis
- Kinematics: finding displacement from velocity and acceleration

##### HL only:

- Implicit differentiation
- Integration by substitution
- Integration by parts

#### Helpful learning resources:

Check out the rest of Khan Academy's limits playlist here.

Check out the rest of Khan Academy's derivatives playlist here.

Check out the rest of Khan Academy's derivative applications playlist here.

Check out the rest of Khan Academy's integrals playlist here.