1. 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)
Summary of Higher
Created by Mr. Lafferty Maths Dept.
Thursday, 10 January 2019

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

21. 4

39. 4

57. 3

77. 2

95. 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

96. 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

97. = + 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

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

99. 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

100. 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

101. 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 -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)
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

102. 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

103. 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) =

104. Special case
Mindmaps

105. 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 + √( ) )
x = cos-1( 1/3)
x = no soln
sinx = ½ + 2√15)
x = 109.5o and 250.5o
Mindmaps

106. 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 =
Amplitude = 2
Mindmaps

107. 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

108. 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

109. 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

110. 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