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# HL analysis and approaches (AA) Homepage

##### Algebra

 Find equations of horizontal and vertical straight lines Understand the the terms gradient and y-intercept. Find equations of diagonal straight lines. 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 term discriminant and how it is used to determine the number of roots of a quadratic. To be able to complete the square for a quadratic. To solve an equation from the completed the square form. Find minimum points and lines of symmetry from graphs of quadratics. Show simple numeric and algebraic proofs. Use of the notation for equality and identity. Write an algebraic fraction in linear terms. Solve simultaneous equations with 3 unknowns using a GDC. Solve simultaneous equations with 3 unknowns using row reduction for a unique solution. Solve simultaneous equations with 3 unknowns using row reduction, showing infinite and no solutions.
##### 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 Understand the terms one-to-one; many-to-one; one-to-many and their applications to functions and inverses. Graph functions with and without a GDC. Note key points on graphs (intersection with axes, maximum and minimum). Understand and use composite functions. Substitute and solve equations using graphs, including composite functions. Using reciprocal graphs, including finding asymptotes from functions with and without sketches. Solve intersection of functions using a GDC. Transformation of functions: reflections in x and y axes; reflection in y=x line; translations; vertical and horizontal stretches. Using the inverse function and the reflection to solve equations. Odd and even functions Sum and roots of a quadratic equation (alpha and beta values). Sum and roots of all polynomial functions, where order is greater than 2. Inequalities of functions and their graphs. The reciprocal of a function and its graph. The modulus of a function and its graph. Factor theorem of polynomials. Remainder theorem of polynomials.
##### 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. Convert between exponent and logarithm form. Use a calculator to solve logarithms. 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). Graphing logarithms and use with transformations and function rules. Modelling with logarithms, such as population models.
##### Series and Expansions
 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 compound interest problems. Understand, use and generate Pascal’s triangle with and without a GDC. Expanding a bracket such as (ax+b)n, where n>2. Problem solving with binomial expansions, including finding terms and constants (independent) terms. Extending binomial expansions for fractional and negative powers. Counting permutations and combinations.
##### 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. Exact trig ratios for 2pi, pi, pi/2, pi/3, pi/4 and pi/6 and multiples of these. Double angle identities for sine and cosine. Circular functions and their periodic nature. Trig graphs and the transformation of trig graphs. Real life trigonometric graphs and modelling. Solving trigonometric equations with and without a GDC. Quadratic equations with trigonometric equations. Reciprocal trigonometric ratios, secx, cosecx, and cotx, and their Pythagorean identities. Inverse trigonometric functions, e.g. arcsinx. Compound angle identities. Double angle identity for tan. Symmetry of trigonometric identities
##### 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 and polynomial equations where the roots are not real. Drawing number in the complex plane, the argand diagram. Writing complex numbers in modulus-argument form. Representing complex numbers in Euler form. Converting between polar-argument and cartesian form with and without technology. Calculations (addition/substraction, multiplication/division) in modulus argument form. Using and proof of de. Moivre’s theorem and using Powers in modulus-argument form.
##### 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. Angles between lines. Kinetic applications of vectors. Intersections of vectors lines, angles, skew and coincident. The vector product rule, and the normal vector. Vector and cartesian equation of a plane. Intersecting planes and link to linear algebra (3×3 solutions). Angles between planes and planes or lines. Intersection points between planes and lines.
##### 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.
##### Reasoning and Proof

 Proof by mathematical induction. Use of proof by contradiction. Use counter-examples to show untrue statements.
##### 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. 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 Finding points of inflexion. Using differentiation for solving optimization problems. Understand informally continuity and differentiability of a function at a point. Differentiate from first principles using the concept of a limit. Use differentiation for higher orders (greater than the second derivative). Evaluation of a limit using Maclaurin’s Series or l’Hopital’s rule, and the repeated use of l’Hopital’s rule. Use implicit differentiation for related rates of change and 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. 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. Using technology to find definite integrals. Finding regions enclosed by two functions. Using integration to solve kinetic problems: displacement, velocity, acceleration. Derivatives of tanx, cosecx, secx, cotx, powers of x, logs, arcsinx, arccosx, arctanx. Use of partial fractions to integrate. Use the substitution rule for integration. Use integration by parts, including repeated integration by parts. Find volumes of revolution for volumes bounded by x and y 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. Homogeneous solutions to differential equations using substitution such as y=vx. Solution of y’+P(x)y=Q(x) using an integrating factor. Applying the MacLaurin’s series for differential equations
##### 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) and variance 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. Continuous random variables and their density functions. Mean, mode, median of continuous random variables. Variance of continuous random variables. Effect of linear transformations on continuous random variables. Using and applying Bayes Theorem.