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# HL applications and interpretations (AI) Homepage

##### Algebra

 Find equations of horizontal and vertical straight lines Understand the the terms gradient and y-intercept. Find equations of diagonal straight lines. Use straight lines to solve 2×2 simultaneous equations. Use a GDC to solve 2×2 simultaneous equations. Factorise simple quadratic equations such as x2+bx+c Factorise simple quadratic equations such as ax2+bx+c where a>1. Solve simple quadratic equations such as x2+bx+c=0 Solve simple quadratic equations such as ax2+bx+c=0 where a>1. Solve quadratic equations by use of a GDC. Understand the quadratic graph, including roots, lines of symmetry and minimum point.

##### Functions

 Understanding the concept of a function. Substitute and solve equations using algebra. Understand the terms domain and range. Find the inverse of a function, including the reflection of a function and its inverse. Graph functions with a GDC. Note key points on graphs (intersection with axes, maximum and minimum). Graphing linear, quadratics and cubic functions. Substitute and solve equations using graphs, including composite functions. Solve intersection of functions using a GDC. Apply knowledge of functions to create and understand models. Understand direct and inverse proportion. Recognise appropriate models and their parameters. Understand piecewise functions. Understand and use composite functions. Substitute and solve equations using graphs, including composite functions. Transformation of functions: reflections in x and y axes; reflection in y=x line; translations; vertical and horizontal stretches.
##### Voronoi Diagrams
 Finding the coordinate of a bisector of a line segment, or between two points. Finding perpendicular line equations. Understanding language of Voronoi diagrams: sites, vertices, edges, cells. Nearest neighbour interpolation. Solving Voronoi problems in context.
##### Number and Logarithms

 Write numbers in a x 10k (standard form). Calculate in standard form. Understand basic rules of exponents such as (multiplying, division, powers, fractional powers, negative powers, etc). Find exponents of integers without a calculator. Use the natural logarithm (ln) and interchange with its base (e). Modelling with logarithms, such as population models. Understanding and calculating simple and compound interest. Using a GDC and other technology to find amortization and annuities. Approximation to decimal places and significant figures. Understanding and finding upper and lower bounds of rounded number. Finding percentage error of rounded number. Estimation of number. Convert between exponent and logarithm form. Understand the basic rules of logarithms and relationship with exponent forms. Understand and use the change of base rule. Use the natural logarithm (ln) and interchange with its base (e). Using logarithm models with large and small numbers. Interpretation of log-log and semi-log graphs.
##### Series

 Understand the term arithmetic sequence. Understand the term geometric sequence. Find the nth term of an arithmetic sequence. Find the sum of n terms of an arithmetic series. Find common differences and problem solve with arithmetic sequence and series. Find the nth term of a geometric sequence. Find the sum of n terms of a geometric series. Find common differences and problem solve with geometric sequence and series. Find the infinite sum of a geometric series. Use the sigma sign for sums of both arithmetic and geometric series. Use series to apply to problem solving in modelling.
##### Trigonometry
 Trigonometric ratios in a right angled triangle. SOHCAHTOA. Sine, cosine rules in non-right angled triangles; ambiguous case of the sine rule. Area of a triangle ½absinC. Using SOHCAHTOA with 3d shapes, including pyramids, cones, spheres, and combinations of these. Using trigonometry in problem solving (Pythagoras’, elevation and depression). Radian and degree measure conversion. Using radian measure, length of an arc and area of a sector. Unit circle and Pythagorean identities, sin2x+cos2x=1, tanx=sinx/cosx. Circular functions and their periodic nature. Trig graphs and the transformation of trig graphs. Real life trigonometric graphs and modelling.
##### Complex Numbers
 Understand the concept of the imaginary number, i, and the powers of i. Calculations with complex numbers in the form a+bi (cartesian form), including addition/substraction, multiplication/division. Finding solutions to quadratic equations where the roots are not real. Drawing number in the complex plane, the argand diagram. Writing complex numbers in modulus-argument form. Converting between polar-argument and cartesian form with and without technology. Calculations (addition/substraction, multiplication/division) in modulus argument form. Powers in modulus-argument form. Understand sinusoidal functions with the same frequencies, but different shift-phases.
##### Vectors
 Concept of a vector in 2d and 3d. Use of vector geometry to show movement, including displacement and position vectors. Ading, subtracting, scalar multiples of vectors. The magnitude of 2d and 3d vectors. Scalar product of two vectors. Finding angles between vectors, including perpendicular and parallel vectors. Vector equations of lines in 3d. Vector applications to motion in 2d and 3d, including kinetics. Geometric interpretation of vectors, such as magnitude of the cross product. Components and direction of vectors.
##### Statistics

 Types of data: continuous and discrete. Sampling: random samples, bias in samples and outliers. Understand how to take a random sample, stratified data, systematic data. Central tendency measure: mean, mode, median, range and quartiles, including grouped data. Understand statistical diagrams such as histograms with even bar width. Reading and constructing cumulative frequency and box and whisker plots. Variance and standard deviation, and effect of constant changes on data. Correlation and regression values, Pearsons’ product moment correlation coefficient. Scatter diagrams, lines of best fit and predicting data from lines. Biased and unbiased estimators
##### Hypothesis Testing
 Calculate Spearman’s Rank coefficient and understand limitations. Analyse outliers and use of Spearman’s rank and Pearson’s correlation on data. Understand the null and alternative hypothesis for testing, the p-value, and significance levels. Conduct and analyse a Chi Squared test for goodness of fit. Conduct and analyse a Chi Squared test for independence. Conduct a t-test to analyse the means of two populations, using one and two tailed data. Design appropriate data collection techniques. Choosing appropriate data to analyse and fitting to relevant data types. Understanding reliability, testing and results with statistics Analysing non-linear regression, and evaluation of least squares regression on data. Sum of square residuals to fit a model. Using and finding the coefficient of determination (R2).
##### Differentiation
 Understand the concept of a limit from a table or graph. Understand that the derivative is a gradient or rate of change function, and its notation. Understand when functions are increasing or decreasing. Simple derivatives by reducing the powers, e.g axn Find the equations of tangents and normals to functions. Derivatives of sinx, cosx, lnx, and ex. Use and understand the chain rule and related rates of change. Use and understand the product and quotient rules. Understand the concept of maximum and minimum points. Find the second derivative, and use this distinguishing between maximum and minimum points Understand the terms concave up and concave down. Using differentiation for solving optimization problems.
##### Integration
 Integrate by using anti-differentiation techniques. Integration and finding the value of the constant, C. Using definite integration in order to find areas under curves. Using technology to find definite integrals. Using the trapezoid rule to estimate areas. Areas enclosed between the x and y axes. Using integration to solve kinetic problems: displacement, velocity, acceleration. Integration techniques to find integrals of sin, cos, ln and x. Integration to find composite functions such as integral of cos(3x+1). Using reverse chain rule to integrate functions. Finding volumes of revolution about the axes.
##### Differential Equations
 Understand and use first order differential equations. Numerical solutions to differential equations using Euler’s method. Separating variables to solve differential equations. Numerical solutions with coupled systems. Phase portrait for the solutions of coupled differential equations. Qualitative analysis of future paths for distinct, real and imaginary eigenvalues. Sketching trajectories and using phase portraits to identify special points. Solutions of the second differential by use of Euler’s method.
##### Probability

 Understand set notation for number, union, intersection, contained in, element, inverse, null and universal. Be able to draw and shade Venn diagrams with the use of set notation. Solve problems by use of sets and Venn diagrams. Understand the the terms mutually exclusive and independent. Find simple probabilities involving one or two events, using the terms AND and OR. Draw tree diagrams to represent probabilities. Use the independence formulae to solve problems of independent probability. Find conditional, or ‘given that’ probabilities. Use Venn diagrams to solve probability problems. Draw and interpret tree diagrams, using them for conditional and independent events. Discrete random variables and their probability distributions. Expected value (mean) for discrete standard variables. Binomial distributions, including mean and variance of these distributions. Normal distribution diagrammatic representation and using this for non-GDC calculations. Using a GDC to find probabilities based on knowing the mean and standard deviation. Finding z-values with normal distributions, and using for inverse calculations. Poisson distributions, mean, variance, and sums of poisson distributions. Confidence intervals for normal distributions with and without a known standard deviation. Test for critical values and regions with normal distributions. Calculating Type I and Type II errors, includes link with Statistics module.
##### Matrices
 Understand definition and order of matrices. Simple operations with matrice: addition, subtraction, scalar multiplying. Multiplication of matrices. Identity and zero matrices and determinants of matrices using technology and by hand. Solving simultaneous equations with matrices and using inverse matrices for solutions. Understanding eigenvalues and eigenvectors, including diagonalization. Powers of 2×2 matrices. Using matrices for transformations. Determinant of a transformation matrix. Transition matrices and powers of transition matrices. Markov chains and initial state probability matrices. Steady and long term probabilities by continued multiplications.

##### Graph Theory
 Definitions of graphs: vertices, edges, degree, adjacency. Draw and understand simple graphs, weighted graphs and connected graphs. Draw and understand trees, subgraphs, directed graphs, degree in and degree out graphs. Understand walks, trials, paths and circuits. Understand Eulerian trails and circuits. Understand Hamiltonian paths and cycles. Use and understand the minimum spanning tree, including Kruskal’s and Prim’s algorithms. Solve Chinese Postman Problems (Eulerian). Using the nearest neighbour algorithm. Using vertex deletion to find lower bounds for the travelling salesman problem (Hamiltonian). Adjacency matrices for walks and graphs. Using weighted adjacency tables for construction of transition matrices.