Chapter 0: Indices, Surds and Logarithms, Quadratic Functions, Remainder and Factor Theorems, Basic Functions and their Graphs

1. Law of Indices

2. Rules of Surds

3. Laws of Logarithms

(change of base formula)

and

4. Roots of Quadratic Equations: 2 real and distinct roots: repeated roots: no real roots

5. Remainder Theorem

When a polynomial is divided by , where , the remainder is

6. Factor Theorem

For a given polynomial , is factor of , where

Chapter 1: Functions, Inverse Functions and Composite Functions

1. Vertical Line TestA relation f is a function if and only if any vertical line , where Domain of f, cuts the graph of f at exactly one point.

2. Horizontal Line TestF is one-one if and only if any horizontal line y = k, where Range of f, cuts the graph of f at exactly one point. (Therefore the inverse function of f exists.)

3. Inverse Functions

4. Composite Functions

To determine if fg exists:

Since , fg exists.

Chapter 2: Graphing Techniques

1. Simple Transformations

Transformation For the equation of the image after transformation

Equation after transformation

Translation in the positive y-direction by a units

Replace y by y – a y = f(x) + a

Translation in the negative y-direction by a units

Replace y by y + a y = f(x) – a

Translation in the positive x-direction by a units

Replace x by x – a y = f(x – a)

Translation in the negative x-direction by a units

Replace x by x + a y = f(x + a)

Stretch parallel to the y-axis by factor a (with x-axis invariant)

Replace y by y/a y = af(x)

Stretch parallel to the x-axis by factor a (with y-axis invariant)

Replace x by x/a y = f(x/a)

Reflection in the x-axis Replace y by -y y = -f(x)Reflection in the y-axis Replace x by -x y = f(-x)

2. Graphs of Rational Functions[IN MF15]

Express in the form of

Vertical asymptote:

Horizontal asymptote:

Express in the form of

Vertical asymptote:

Horizontal asymptote:

3. Modulus Functions

y = |f(x)| Keep part of the graph of y = f(x) for which f(x) 0Reflect the part of the graph of y = f(x) for which f(x) < 0 about the x-axis

y = f(|x|) Keep the part of the graph of y = f(x) for which x 0Remove the part of the graph of y = f(x) for which x < 0Reflect the part of the graph of y = f(x) for which x > 0 about the y-axis

4. Circle(x – a)2 + (y – b)2 = r2, where r > 0Mark center and radius, x and y-axis must have the same scale, note the position of the circle with respect to the origin

5. Ellipse

, where h,k > 0

Symmetrical about lines x = a and y = bScale of both axes should be the sameMark center, “horizontal” and “vertical” radius

6. Parabola (y – a)2 = k (x – b), (x – a)2 = k (y – b) where k 0Mark line of symmetry: y = a and x = a respectivelyMark turning point

7. Hyperbola

, where h,k > 0

Symmetrical about x-axis and y-axisMark two oblique asymptotes and intercepts

Chapter 3: Equations and Inequalities

1. Rules of Inequalities a > b and c > 0 ac > bc a > b and c < 0 ac < bc

> 0 ab > 0 < 0 ab < 0

Chapter 4a: Differentiation I – Techniques of Differentiation and Limits

1. Important Results and Rules

If y = c where c is a constant, then

If y = mx where m is a constant, then

If where n is a constant, then

Product rule:

Quotient rule:

Chain rule: If y is a function of u and u is a function of x, then

2. Exponential and Logarithm Functions

3. Trigonometric Functions

*Above results hold for x measured in radians only. If x is in degrees, then convert to

radian

4. Inverse Trigonometric Functions

5. Higher Order Derivatives

6. Implicit Differentiation

7. Parametric Differentiation

If x = f(t) and y = g(t), then

Chapter 4b: Differentiation II – Applications of Differentiation

1. Strictly Increasing and Strictly Decreasing

Strictly increasing:

Strictly decreasing:

2. Concave Upwards and Concave Downwards

Concave upwards:

Concave downwards:

3. Determining Stationary Points

x a- a a+ a- a a+ a- a a+

-ve 0 -veor

+ve 0 +ve

+ve 0 -ve -ve 0 +ve

Tangent to curve

or

Nature of stationary point

Stationary point of inflexion

Maximum turning point

Minimum turning point

If , then (a, f(a)) is a maximum turning point

If , then (a, f(a)) is a minimum turning point

4. Graph of f’(x)Stationary points on f(x) y = 0 on f’(x)Maximum / minimum gradient on f(x) maximum / minimum points on f’(x)

5. Graph of

y = f(x)

x-intercept (vertical asymptote) at x = h vertical asymptote (x-intercept) at x = hy-intercept at y = k, k 0

y-intercept at y = , k 0

maximum (minimum) point at (a,b), b 0minimum (maximum) point at , b 0

horizontal asymptote at y = q, where q 0horizontal asymptote at , where q 0

f(x) 0

f(x) = 1 = 1

f(x) > 0 [f(x) < 0] > 0 [f(x) < 0]

increasing (decreasing) in (a,b) decreasing (increasing) in (a,b)

6. Graph of Sketch y = f(x)

Consider f(x) 0. Square root the y-values to obtain

Reflect in the x-axis to obtain

7. Relationship between Gradients of Tangent and Normal to a CurveIf A (a, f(a)) is a point on the graph of y = f(x), then The gradient of the tangent at A is f’(a), and

The gradient of the normal at A is

8. Equations of Tangents and Normals to a CurveIf A (a, f(a)) is a point on the graph of y = f(x), then Equation of tangent to the curve at A is

o y – f(a) = f’(a)(x – a) Equation of the normal to the curve at A is

o y – f(a) = (x – a), provided

9. Connected Rates of Change Denote each changing quantity by a variable Find the equations relating the variables Use the chain rule to link up the derivatives Write down the values for the variables and rates given Solve for the unknown rate

Chapter 5a: Integration Techniques

1. Basic Rules of Indefinite Integral[f(x) g(x)] dx = f(x) dx g(x) dxkf(x) dx = k f(x) dx, where k is a constant, k 0

= f(x) + c

= f(x)

2. Basic Properties of Definite Integral

, where c is such that

, where k is any constant,

3. Integration of Standard Functions

[IN MF15]

[IN MF15]

4. Trigonometric Formulae

5. Partial Fractions [IN MF15]

6. Substitution

7. By Parts

Choose u based on LIATE

Chapter 5b: Applications of Integration

1. Fundamental Theorem of Calculus

2. Parametric Equations

3. Volume of Solid of Revolution

Rotation about x-axis:

Rotation about y-axis:

Chapter 7: Arithmetic and Geometric Progression

1. Arithmetic Progression

2. Geometric Progression

Chapter 8: Summation of Series and Mathematical Induction

1. Properties of the Σ Notation

2. Standard Results

Sum of AP:

Sum of GP:

*must always start from r=1 [IN MF15]

*must always start from r=1 [IN MF15]

3. Mathematical Induction Let be the statement “…”

Show that is true

Prove that the implication is true

o Assume is true, prove that is true

o is true whenever is true

Since is true, by Mathematical Induction, is true for

Chapter 10: Binomial Expansion

1. Binomial Expansion for Positive Integral Index

For , , where

[IN MF15]

term:

2. Binomial Expansion for Rational Index

[IN MF15]

*Condition: |x| <1

Chapter 11: Power Series

1. Maclaurin’s Theorem

[IN MF15]

2. Small Angle ApproximationsFor all small angles, positive or negative, we have

where x is measured in radians

Chapter 12A: Vectors I

1. Vector AlgebraTriangle:

Parallelogram:

Polygon:

2. Fundamental ResultsParallel vectors: for some

Unit vectors:

Collinearity: , with A as common pointNon-parallel and non-zero vectors: for some , then

Ratio theorem: If , then . If P is the mid-point of AB, then

.

QP

RS

T

P

A

OB

a

r

b

3. Properties of VectorsCartesian form:

Matrix form:

Equality of vectors:

Addition:

Subtraction:

Multiplication: for any

Distance between two points:

4. Scalar Dot Product

(considering acute angle)

if and if

or

Perpendicular vectors:

Length of projection:

*Cosine rule:

A

O BP

a

b

5. Equation of a Straight Line

Vector form:

Parametric form:

Cartesian form:

If =0, line is parallel to yz-plane.

Determining if point lies on line: equate point to equation of line and see whether able to solve for unique value of lamda

Determining foot of perpendicular: Foot lies on horizontal line Vertical line which the point lies on DOT direction vector of horizontal line = 0 Solve simultaneous equations

Perpendicular distance from point to line Find length of projection to line. Then apply Pythagoras’ Theorem Find foot of perpendicular from point to line. Then find distance.

Characteristics of a pair of lines Parallel: direction vector of one line multiple of the other line Not parallel but intersect: unique value of lamda and miu that satisfies the

equations of both lines when . Once the unique value is found, can substitute it into the equation of one line to find the point of intersection

Not parallel and do not intersect: no unique value

Angle between two lines:

Chapter 12b: Vectors II

1. Vector Product

Parallel:

Perpendicular:

Cartesian notation:

*Area of triangle (sine rule):

*Area of parallelogram:

Perpendicular distance from point to line:

2. Equation of a Plane

Vector form:

Scalar product form: Cartesian form:

Foot of perpendicular from point to plane: Foot lies on equation of normal of plane Foot lies on plane Solve simultaneous equations

Perpendicular distance from point to plane:

A

C BK

a

b

Characteristics between line and plane: Parallel and lies on plane: direction vector of line DOT normal of plane = 0 +

point on line fits the equation of the plane Parallel but does not lie on plane: direction vector of line DOT normal of plane =

0 + point on line does not fit the equation of the plane Not parallel intersects at a point: direction vector of line DOT normal of plane

does not = 0o To find point of intersection: point lies on equation of line and plane. So

equate both equations together and solve for lamda and subsequently the point of intersection

Angle between line and plane:

Angle between two planes:

Characteristics of two planes: Parallel: normal vector of one plane is a multiple of normal vector of the other

plane Intersect to yield common line of intersection: Let z = t. Then solve simultaneous

equations of both equations of planes

Characteristics of three planes: Intersect at single point: unique solution for x, y and z Intersect along one common line: solution for x and y expressed in terms of z Do not intersect at common point or line: no solution found

Chapter 13a: Complex Numbers I

1.Properties of Conjugate Pairs

Important:

Important:

z is real

2. Properties of Complex Conjugates

3. Forms of Complex NumbersCartesian form: Polar / Trigonometric form:

Exponential form:

4. Geometrical Effect of Multiplying 2 Complex NumbersIf P represents the point with complex number z in an Argand diagram, the effect of multiplying z by or is to rotate the line segment of OP through an angle of about O in the anti-clockwise direction followed by a scaling factor r.

When you multiply a complex number by i, you are shifting it radians about the

origin.

5. De Moivre’s TheoremIf , then

If , then Need to add to the angle to get complex roots

Chapter 13b: Complex Numbers II

1. Circle

Locus is a circle with center a and radius r.Indicate intercepts if obvious, indicate radius, check if circle passes through origin

2. Perpendicular Bisector

Locus is the perpendicular bisector of the line segment joining the points a and b.Draw perpendicular bisector with solid line but draw the line adjoining both points with dashed line.Show the perpendicular sign and indicate that the line adjoining both points have been segmented exactly in half.Label intercepts after getting Cartesian equation (if question asks you to determine this)Check whether line passes through origin

3. Half-line

Locus is a half-line starting at (but not including) a, and making an angle of with the positive real axis.Remember to indicate that the starting point is exclusive (ie. draw a circle where the half-line starts), indicate argument, use dashed lines to represent positive real axisFor Cartesian equation, must indicate the domain of x (eg. x > 0)Check whether line passes through origin