New Higher Past Papers by Topic 2015 Straight Line (4) Trigonometry (3) Composite Functions (3) The Circle (4) Differentiation (5) Vectors (4) www.mathsrevision.com Recurrence Relations (2) Logs & Exponential (3) Polynomials (5) Wave Function (2) Integration (5) Summary of Higher Created by Mr. Lafferty Maths Dept. Thursday, 10 January 2019

Higher Mindmaps by Topic Straight Line Trigonometry Composite Functions The Circle Differentiation Vectors www.mathsrevision.com Recurrence Relations Logs & Exponential Polynomials Wave Function Integration Main Menu Thursday, 10 January 2019 Created by Mr. Lafferty Maths Dept.

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Parallel lines have same gradient Distance between 2 points m < 0 m = undefined Terminology Median – midpoint Bisector – midpoint Perpendicular – Right Angled Altitude – right angled m1.m2 = -1 m > 0 m = 0 Possible values for gradient Form for finding line equation y – b = m(x - a) (a,b) = point on line Straight Line y = mx + c Parallel lines have same gradient m = gradient c = y intercept (0,c) For Perpendicular lines the following is true. m1.m2 = -1 θ m = tan θ Mindmaps

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

= + x x g(x) = g(f(x)) f(x) = x2 - 4 g(f(x)) = y = f(x) g(x) = f(g(x)) 1 x g(f(x)) f(x) = x2 - 4 But y = f(x) is x2 - 4 1 x2 - 4 g(f(x)) = 1 y x y = f(x) Restriction Domain Range x2 - 4 ≠ 0 A complex function made up of 2 or more simpler functions Similar to composite Area (x – 2)(x + 2) ≠ 0 x ≠ 2 x ≠ -2 Composite Functions 1 x = + But y = g(x) is g(x) = 1 x f(g(x)) 1 x 2 - 4 f(x) = x2 - 4 f(g(x)) = x 1 x2 - 4 y = g(x) y2 - 4 Rearranging Restriction x2 ≠ 0 Domain Range Mindmaps

Basics before Differentiation/ Integration Format for Differentiation / Integration Surds Indices Basics before Differentiation/ Integration Division Working with fractions Adding Multiplication Subtracting Mindmaps

Equation of tangent line Nature Table Equation of tangent line Leibniz Notation -1 2 5 + - x f’(x) Max Straight Line Theory Gradient at a point f’(x)=0 Stationary Pts Max. / Mini Pts Inflection Pt Graphs f’(x)=0 Derivative = gradient = rate of change Differentiation of Polynomials f(x) = axn then f’x) = anxn-1 Mindmaps

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

f(x) =2x2 + 4x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 Completing the square f(x) = a(x + b)2 + c Easy to graph functions & graphs 1 4 5 2 -2 -2 -4 -2 Factor Theorem x = a is a factor of f(x) if f(a) = 0 1 2 1 f(x) =2x2 + 4x + 3 f(x) =2(x + 1)2 - 2 + 3 f(x) =2(x + 1)2 + 1 (x+2) is a factor since no remainder If finding coefficients Sim. Equations Discriminant of a quadratic is b2 -4ac Polynomials Functions of the type f(x) = 3x4 + 2x3 + 2x +x + 5 Degree of a polynomial = highest power Tangency b2 -4ac > 0 Real and distinct roots b2 -4ac < 0 No real roots b2 -4ac = 0 Equal roots Mindmaps

b (1 - a) Un+1 = aUn + b a > 1 then growth a < 1 then decay Limit L is equal to Given three value in a sequence e.g. U10 , U11 , U12 we can work out recurrence relation U11 = aU10 + b (1 - a) L = b U12 = aU11 + 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 = aUn + b |a| > 1 |a| < 1 a > 1 then growth a < 1 then decay Limit exists when |a| < 1 + b = increase - b = decrease Mindmaps

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

Special case Mindmaps

C A S T 0o 180o 270o 90o xo 4 2 The exact value of sinx Trig Formulae Double Angle Formulae sin2A = 2sinAcosA cos2A = 2cos2A - 1 = 1 - 2sin2A = cos2A – sin2A Addition Formulae sin(A ± B) = sinAcosB cosAsinB cos(A ± B) = cosAcosB sinAsinB C A S T 0o 180o 270o 90o xo 4 2 The exact value of sinx Trig Formulae and Trig equations 3cos2x – 5cosx – 2 = 0 Let p = cosx 3p2 – 5p - 2 = 0 (3p + 1)(p -2) = 0 sinx = 2sin(x/2)cos(x/2) p = cosx = 1/3 cosx = 2 sinx = 2 (¼ + √(42 - 12) ) x = cos-1( 1/3) x = no soln sinx = ½ + 2√15) x = 109.5o and 250.5o Mindmaps

Basic Strategy for Solving Rearrange into sin = Find solution in Basic Quads Remember Multiple solutions Exact Value Table C A S T 0o 180o 270o 90o ÷180 then X π degrees radians Basic Strategy for Solving Trig Equations ÷ π then x 180 Trigonometry sin, cos , tan 360o 1 -1 Amplitude Complex Graph 90o 2 -1 1 180o 270o 360o 3 Period sin x Basic Graphs Amplitude 1 -1 360o Amplitude 1 -1 180o 90o Period Period y = 2sin(4x + 45o) + 1 Period cos x Max. Value =2+1= 3 Period = 360 ÷4 = 90o tan x Mini. Value = -2+1 -1 Amplitude = 2 Mindmaps

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

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

y = logax y = ax a0 = 1 a1 = a y = axb y = abx To undo log take exponential y = ax y x To undo exponential take log y x (a,1) loga1 = 0 (1,a) a0 = 1 (0,1) logaa = 1 (1,0) a1 = a Basic log graph Basic exponential graph log A + log B = log AB log A - log B = log B A log (A)n = n log A Logs & Exponentials Basic log rules y = axb Can be transformed into a graph of the form y = abx Can be transformed into a graph of the form (0,C) log y log x (0,C) log y x log y = x log b + log a log y = b log x + log a Y = mX + C Y = mX + C Y = (log b) X + C Y = bX + C C = log a m = log b C = log a m = b Mindmaps

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