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Additional Mathematics

0606

Understand the terms function, domain, range (image set), one–one function, many–one function, inverse function and composition of functions Find the domain and range of functions, including inverse and composite functions Recognise and use function notation such as f(x), f⁻¹(x), fg(x) and f²(x) Understand the relationship between y = f(x) and y = |f(x)| for linear, quadratic, cubic or trigonometric functions Explain in words why a given function does not have an inverse Find the inverse of a one–one function using correct notation Form and use composite functions, understanding that the order of composition is important Use sketch graphs to show the relationship between a function and its inverse as reflections in the line y = x Find the maximum or minimum value of a quadratic function by completing the square or by differentiation Use the maximum or minimum value of a quadratic function to sketch its graph or to determine the range for a given domain Know the conditions for a quadratic equation to have two real roots, two equal roots or no real roots and the related conditions for a line to intersect, be tangent to or not intersect a curve Solve quadratic equations for real roots using factorisation, the quadratic formula or completing the square Find the solution set for quadratic inequalities either graphically or algebraically, expressing solutions using correct inequality notation Know and use the remainder and factor theorems Find factors of polynomials, including factorising a cubic into a linear factor and a quadratic factor Solve cubic equations using factorisation or other appropriate methods Solve equations involving moduli such as |ax + b| = c, |ax + b| = cx + d, |ax + b| = |cx + d| and |ax² + bx + c| = d using algebraic or graphical methods Solve graphically or algebraically inequalities involving moduli, including forms such as k|ax + b| > c, k|ax + b| ≤ c, k|ax + b| ≤ |cx + d|, |ax + b| ≤ cx + d and |ax² + bx + c| > or ≤ d Use substitution to form and solve a quadratic equation in order to solve a related equation Sketch graphs of cubic polynomials and their moduli when given as a product of three linear factors, clearly showing intercepts Solve graphically cubic inequalities of the form f(x) ≥ d, f(x) > d, f(x) ≤ d or f(x) < d where f(x) is a product of three linear factors Solve simultaneous equations in two unknowns by elimination or substitution, including equations that are linear and non-linear Know and use simple properties and graphs of logarithmic and exponential functions, including ln x and e^x, and understand their inverse relationship and asymptotes Know and use the laws of logarithms, including change of base, and express combinations of logarithms as a single logarithm Solve equations of the form a^x = b using logarithms or other appropriate methods Use the equation of a straight line to solve problems involving gradients and intercepts Know and use the condition for two lines to be parallel or perpendicular in terms of their gradients Solve problems involving midpoint and length of a line segment and find and use the equation of a perpendicular bisector Transform given relationships to and from straight-line form and determine unknown constants by calculating gradient or intercept of the transformed graph Know and use the equation of a circle with radius r and centre (a, b) and identify the centre and radius from a circle equation in various forms Solve problems involving the intersection of a circle and a straight line, including deciding whether the line is a tangent, a chord or does not meet the circle Solve problems involving tangents to a circle, including finding equations of tangents Solve problems involving the intersection of two circles, including finding points of intersection, the equation of a common chord and deciding whether two circles intersect, touch or do not meet Solve problems involving arc length and sector area of a circle using radian measure, including compound shapes Know and use the six trigonometric functions of angles of any magnitude: sine, cosine, tangent, secant, cosecant and cotangent Understand and use the amplitude and period of a trigonometric function and the relationships between graphs of related trigonometric functions Draw and use graphs of y = a sin(bx) + c, y = a cos(bx) + c and y = a tan(bx) + c over a given domain, including identification of asymptotes for tangent graphs Use the identities sin²A + cos²A = 1, sec²A = 1 + tan²A and cosec²A = 1 + cot²A Solve, for a given domain, trigonometric equations involving any of the six trigonometric functions, possibly using standard identities Prove trigonometric relationships involving the six trigonometric functions using standard identities Recognise the difference between permutations and combinations and know when each should be used Know and use the notation n! and the standard expressions for permutations and combinations of n items taken r at a time Solve arrangement and selection problems in context using permutations or combinations Use the binomial theorem for the expansion of (a + b)^n for positive integer n, including simplification of coefficients Use the general term in the binomial expansion to find a specified term or to solve related problems Recognise arithmetic and geometric progressions and understand the difference between them Use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions Use the condition for convergence of a geometric progression and the formula for the sum to infinity of a convergent geometric progression Understand and use vector notation in various forms, including column vectors and directed line segments Know and use position vectors and unit vectors, including forming a unit vector parallel to a given vector Find the magnitude of a vector and add, subtract and multiply vectors by scalars, including solving vector geometry problems Compose and resolve velocities, determine resultant vectors and use velocity vectors to solve problems in context such as motion and collisions Understand the idea of a derived function as the rate of change and the link with gradients of curves Use standard differentiation notation including first and second derivatives Know and use derivatives of standard functions x^n for rational n, sin x, cos x, tan x, e^x and ln x, including constant multiples, sums and simple composite functions using the chain rule Differentiate products and quotients of functions using the product and quotient rules Use differentiation to find gradients, tangents and normals to curves Use differentiation to find stationary points of functions Apply differentiation to connected rates of change, small increments and approximations Apply differentiation to practical problems involving maxima and minima in context Use first and second derivative tests to distinguish between maxima and minima and justify conclusions Understand integration as the reverse process of differentiation and include an arbitrary constant in indefinite integrals Integrate sums of terms in powers of x, including x⁻¹ and 1/(ax + b), including an arbitrary constant Integrate functions of the form (ax + b)^n, sin(ax + b), cos(ax + b), sec²(ax + b) and e^(ax + b), including the case n = –1 Evaluate definite integrals and apply integration to find plane areas between curves and lines Apply differentiation and integration to kinematics of a particle moving in a straight line, relating displacement, velocity and acceleration for constant or variable acceleration Use displacement–time, distance–time, velocity–time, speed–time and acceleration–time graphs to interpret and solve kinematics problems

Use the identities sin²A + cos²A = 1, sec²A = 1 + tan²A and cosec²A = 1 + cot²A

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