www mathsrevision com New Higher Past Papers by

www. mathsrevision. com New Higher Past Papers by Topic 2015 Straight Line (4) Trigonometry (3) Composite Functions (3) The Circle (4) Differentiation (5) Vectors (4) Recurrence Relations (2) Logs & Exponential (3) Polynomials (5) Wave Function (2) Integration (5) Wednesday, December 15, 2021 Summary of Higher Created by Mr. Lafferty Maths Dept.

www. mathsrevision. com Higher Mindmaps by Topic Straight Line Trigonometry Composite Functions The Circle Differentiation Vectors Recurrence Relations Logs & Exponential Polynomials Wave Function Integration Wednesday, December 15, 2021 Main Menu Created by Mr. Lafferty Maths Dept.

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m<0 m = undefined m=0 m>0 Possible values for gradient Straight Line y = mx + c Parallel lines have same gradient For Perpendicular lines the following is true. m 1. m 2 = -1 m = tan θ Mindmaps θ

+ flip in y-axis Move vertically up or downs depending on k - Stretch or compress vertically depending on k y = f(x) ± k y = f(-x) Remember we can combine these together !! y = kf(x) Graphs & Functions y = -f(x) y = f(kx) y = f(x ± k) flip in x-axis - + Move horizontally left or right depending on k Stretch or compress horizontally depending on k Mindmaps

f(x) = x 2 1 g(x) = x -4 x g(f(x)) 1 y y = f(x) Domain But y = f(x) is x 2 - 4 g(f(x)) = Range A complex function made up of 2 or more simpler functions Similar to composite Area = 1 g(x) = x x Domain f(x) = y = g(x) x 2 x ≠ 2 - 4 f(g(x)) y 2 - 4 Range x 2 - 4 ≠ 0 (x – 2)(x + 2) ≠ 0 Composite Functions + Restriction x ≠ -2 1 x But y = g(x) is f(g(x)) = 1 x 2 -4 Rearranging Restriction x 2 ≠ 0 Mindmaps

Format for Differentiation / Integration Surds Indices Basics before Differentiation/ Integration Working with fractions Mindmaps

Nature Table Equation of tangent line Leibniz Notation Gradient at a point f’(x)=0 Stationary Pts Max. / Mini Pts Inflection Pt Graphs f’(x)=0 Differentiation of Polynomials f(x) = axn then f’x) = anxn-1 Mindmaps Straight Line Theory Derivative = gradient = rate of change

Trig Harder functions Use Chain Rules of Indices Polynomials Differentiations Real life Graphs Meaning Stationary Pts Mini / Max Pts Inflection Pts Rate of change of a function. Gradient at a point. Factorisation Tangent equation Mindmaps Straight line Theory

Completing the square f(x) = a(x + b)2 + c Easy to graph functions & graphs Factor Theorem x = a is a factor of f(x) if f(a) = 0 f(x) =2 x 2 + 4 x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 -2 1 4 5 2 -2 -4 -2 1 0 (x+2) is a factor since no remainder If finding coefficients Sim. Equations Discriminant of a quadratic is b 2 -4 ac Polynomials Functions of the type f(x) = 3 x 4 + 2 x 3 + 2 x +x + 5 Tangency b 2 -4 ac > 0 Real and distinct roots b 2 -4 ac = 0 Equal roots Degree of a polynomial = highest power b 2 -4 ac < 0 No real roots Mindmaps

Limit L is equal to U 11 = a. U 10 + b Given three value in a sequence e. g. U 10 , U 11 , U 12 we can work out recurrence relation U 12 = a. U 11 + b Use Sim. Equations a = sets limit b = moves limit Un = no effect on limit Recurrence Relations next number depends on the previous number Un+1 = a. Un + |a | > 1 Limit exists when |a| < 1 b |a | < 1 a > 1 then growth a < 1 then decay + b = increase - b = decrease Mindmaps

f(x) g(x) Remember to change sign to + if area is below axis. b A= ∫a f(x) - g(x) dx Finding where curve and line intersect f(x)=g(x) gives the limits a and b Area between 2 curves Integration of Polynomials IF f’(x) = axn Then I = f(x) = Remember to work out separately the area above and below the x-axis. Integration is the process of finding the AREA under a curve and the x-axis

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Double Angle Formulae sin 2 A = 2 sin. Acos. A cos 2 A = 2 cos 2 A - 1 = 1 - 2 sin 2 A = cos 2 A – sin 2 A Addition Formulae sin(A ± B) = sin. Acos. B cos. Asin. B cos(A ± B) = cos. Acos. B sin. Asin. B Trig Formulae and Trig equations 3 cos 2 x – 5 cosx – 2 = 0 Let p = cosx 3 p 2 – 5 p - 2 = 0 sinx = 2 (¼ + √(42 - 12) ) (3 p + 1)(p -2) = 0 cosx = 2 p = cosx = 1/3 x = no soln x = cos-1( 1/3) sinx = ½ + 2√ 15) x = 109. 5 o and 250. 5 o sinx = 2 sin(x/2)cos(x/2) Mindmaps

1. 2. 3. Rearrange into sin = Find solution in Basic Quads Remember Multiple solutions Basic Strategy for Solving Trig Equations Trigonometry sin, cos , tan Amplitude Complex Graph Period sin x Basic Graphs Amplitude Period cos x Amplitude Period tan x y = 2 sin(4 x + 45 o) + 1 Max. Value =2+1= 3 Period = 360 ÷ 4 = 90 o Mini. Value = -2+1 -1 Amplitude = 2 Mindmaps

same for subtraction Addition 2 vectors perpendicular if Scalar product Component form Magnitude Basic properties Q B P A a Vectors are equal if they have the same magnitude & direction scalar product Vector Theory Magnitude & Direction Notation Component form Unit vector form Mindmaps

b Tail to tail θ a Angle between two vectors properties C Vector Theory Magnitude & Direction Section formula B B A m Points A, B and C are said to be Collinear if A B is a point in common. Mindmaps C n c b a O

y y = logax (a, 1) To undo log take exponential loga 1 = 0 logaa = 1 (1, 0) x log A + log B = log AB To undo exponential take log Basic log graph log A - log B = log A B n log (A) = n log A Basic log rules y= Basic exponential graph Logs & Exponentials y = axb Can be transformed into a graph of the form log y = x log b + log a (0, C) x a 0 = 1 a 1 = a (1, a) x abx log y y (0, 1) y = ax Y = m. X + C Y = (log b) X + C C = log a m = log b log y = b log x + log a Y = m. X + C Y = b. X + C C = log a m=b log y (0, C) log x Mindmaps

f(x) = a sinx + b cosx Compare coefficients compare to required trigonometric identity a = k cos β f(x) = k sin(x + β) = k sinx cos β + k cosx sin β b = k sin β Process example Square and add then square root gives Divide and inverse tan gives Wave Function a and b values decide which quadrant transforms f(x)= a sinx + b cosx Write out required form into the form OR Related topic Solving trig equations Mindmaps