IB Mathematics SL

Algebra

Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Sigma notation. Elementary treatment of exponents and logarithms. Laws of exponents; laws of logarithms. Change of base.

The binomial theorem.




Functions and equations

Domain, range, image of a function. Composite functions. Identity function. Inverse function.

The graph of a function, Function graphing skills., Investigation of key features of graphs, such as

maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.

Use of technology to graph a variety of functions, including ones not specifically mentioned.

The graph of inverse function as the reflection in the line y = x of the graph of y = f (x) .

Transformations of graphs , Translations, Reflections, Vertical stretch with scale factor p, Stretch in the x-direction, Composite transformations.

The quadratic function, The reciprocal function, The rational function, Vertical and horizontal asymptotes.




Exponential functions and their graphs, Logarithmic functions and their graphs.

Solving equations, both graphically and analytically.

Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.

The quadratic formula, The discriminant, Solving exponential equations, Applications of graphing skills and solving

equations that relate to real-life situations.




Circular functions and trigonometry

The circle: radian measure of angles; length of an arc; area of a sector. Definition of cosθ and sinθ in terms of the

unit circle. Exact values of sin, cos and tan of and their multiples.

The Pythagorean identity, Double angle identities for sine and cosine, Relationship between trigonometric ratios.

The circular functions sin x , cos x and tan x :their domains and ranges; amplitude, theirperiodic nature; and their graphs.

Composite functions of the form f (x) = asin (b(x + c) ) + d . Transformations.

Solving trigonometric equations in a finite interval, both graphically and analytically.

Equations leading to quadratic equations in sin x, cos x or tan x .

Solution of triangles, The cosine rule, The sine rule, including the ambiguous case, Area of a triangle.




Vectors

Vectors as displacements in the plane and in three dimensions.

Components of a vector, The scalar product of two vectors, Perpendicular vectors; parallel vectors, The angle between two vectors,

Vector equation of a line in two and three dimensions, The angle between two lines, Distinguishing between coincident and parallel

lines. Finding the point of intersection of two lines. Determining whether two lines intersect.




Statistics and probability

Concepts of population, sample, random sample, discrete and continuous data.

Presentation of data: frequency distributions, frequency histograms with equal class intervals, box-and-whisker plots, outliers.

Grouped data: use of mid-interval values for calculations, interval width; upper and lower interval boundaries, modal class.

Statistical measures and their interpretations. Central tendency: mean, median, mode. Quartiles, percentiles.

Dispersion: range, interquartile range, variance, standard deviation. Effect of constant changes to the original data.

Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles.

Linear correlation of bivariate data, Pearson’s product–moment correlation coefficient r.

Scatter diagrams; lines of best fit., Equation of the regression line of y on x. Use of the equation for prediction purposes.

Mathematical and contextual interpretation. Concepts of trial, outcome, equally likely

outcomes, sample space (U) and event. The probability of an event A, The complementary events A and A′ (not A).

Use of Venn diagrams, tree diagrams and tables of outcomes.

Combined events, Mutually exclusive events, Conditional probability, Independent events, Probabilities with and without replacement.

Concept of discrete random variables and their probability distributions. Expected value (mean), E(X ) for discrete data.

Binomial distribution. Mean and variance of the binomial distribution. Normal distributions and curves. Standardization of normal variables (z-values, z-scores). Properties of the normal distribution.




Calculus

Informal ideas of limit and convergence, Limit notation, Definition of derivative from first principles, Derivative interpreted as gradient function and as rate of change. Tangents and normals, and their equations. The chain rule for composite functions.

The product and quotient rules. The second derivative. Extension to higher derivatives. Local maximum and minimum points.

Testing for maximum or minimum. Points of inflexion with zero and non-zero gradients. Graphical behaviour of functions,

including the relationship between the graphs of f , f ′ and f ′′ . Optimization. Indefinite integration as anti-differentiation.

The composites of any of these with the linear function ax + b . Integration by inspection, or substitution.

Anti-differentiation with a boundary condition to determine the constant term. Definite integrals, both analytically and using

technology. Areas under curves (between the curve and the x-axis). Areas between curves. Volumes of revolution about the x-axis.

Kinematic problems involving displacement s, velocity v and acceleration a. Total distance travelled.