Pure Year 1 Overview

Proof

Logical connectives

The three logical connectives are:

If P, then Q ()
If and only if P, then Q ()
P is implied by Q () (not in spec)

Disproof by counter example

Disproof by counter example involves disproving an "if then " statement by finding an example where is true but is not.

Identities

An identity is a relation that is true for all values of the variable. An identity uses the symbol , and the two sides of the identity are congruent expressions.

Sets of numbers

Integers

- whole numbers.

Real numbers

- any number on the real number line, including all rational and irrational numbers.

Rational numbers

- any number than can be represented as , where p and q are integers, and is nonzero.

Irrational numbers

Any real number that is not rational, e.g. or .

Inequality notation

Interval notation

A round bracket or means the number is not included, e.g. is the interval from 3 to 5, non-inclusive. A square bracket or means the number is included, e.g. means the interval from 3 to 5, inclusive.

Equivalent interval notation

:
:
:
:
:
:
or :

Indices

Laws of indices

Multiplication:
Division:
Exponentiation:
Fractional exponents:
Negative exponents

Rationalising the denominator

To rationalise the denominator of a fraction, multiply by the conjugate radical of the denominator (e.g. for ) to create a difference of two squares.

Quadratics

Quadratic formula

Derivation

The quadratic formula can be derived by completing the square on a general quadratic .

Discriminant

The discriminant of a quadratic equation is .

Meaning

has two distinct real roots.
has one repeated real root.
has no real roots.

Line of symmetry

The line of symmetry is at .

Polynomials

Factor theorem

is a factor of if and only if .

Graph sketching

Consider:

The basic shape based on the degree and the lead coefficient.
The y-intercept.
The x-intercepts (roots).
How the curve meets each root (e.g. repeated roots, point of inflection).

Graph transformations

Translation vertically by

Translation horizontally by

Vertical stretch, scale factor a, relative to the -axis

Horizontal stretch, scale factor , relative to the -axis

Reflection in the -axis

Coordinate geometry

Straight line equations

Midpoint of two points

Distance between two points

Perpendicular lines

Perpendicular lines have gradients that multiply to -1.

Circle equation

for a circle centred at with radius .

Circle theorems

The angle in a semicircle is a right angle.
The perpendicular from the centre of a circle to a chord bisects the chord.
The radius of a circle is perpendicular to a tangent at the circumference.

Intersections of circles

There are 5 cases (where ):

: The two circles do not intersect, with the smaller circle outside the larger circle.
: The two circles intersect in one place, where they are tangent, with the smaller circle outside the larger circle (externally tangent).
: The two circles intersect in two places.
: The two circles intersect in one place, where they are tangent, with the smaller circle inside the larger circle (internally tangent).
: The two circles do not intersect, with the smaller circle inside the larger circle.

Logarithms

Rules of logarithms

Converting:
Multiplication
Division:
Exponentiation:

Specific bases

Base

Logarithms in base are written as .

Base 10

Logarithms in base 10 are written as .

Graph sketching

A logarithmic function of the form has:

An -intercept of
The -axis as an asymptote.

Exponentials

Graph sketching

An exponential of the form has:

A -intercept of
All
The -axis as an asymptote.

Growth and decay

For an exponential of the form :

If , then increases as increases - exponential growth.
If , then decreases as increases - exponential decay.

Modelling

Exponential functions are used to model situations where the rate of growth or decay is proportional to the current amount. This is because the derivative of is .

Equation

A model with equation has:

Initial value () of
A rate of change of .

Straightening graphs

Exponential

For a graph , taking on both sides gives:

Thus when plotting against , the gradient is and -intercept is .

Polynomial

For a graph , taking on both sides gives:

Thus when plotting against , the gradient is and the -intercept is .

Binomial expansion

The expansion of can be found by using the expansion:

Binomial coefficient

The binomial coefficient, written or is given by:

Trigonometry

Trig identities

Sine rule

When using the sine rule to find angles, sometimes there may be two possible answers, and .

Cosine rule

Area of a triangle

Vectors

A vector has both magnitude and direction.

Magnitude

The magnitude of a vector is denoted and can be found using the Pythagorean theorem.

Direction

The direction of a vector is measured anticlockwise from the positive axis and can be found using .

Unit vector

A unit vector has a magnitude of 1.

Operations

Vectors can be, added, subtracted, and multiplied by a scalar.

Parallel vectors

If vectors and are parallel then for some scalar , .

Position vectors

Vectors can represent positions of points or the displacement between two points. The position vector of a point is a vector from the origin to that point.

Displacement vector

For two points and with position vectors and , the displacement vector from to , , is given by .

Distance

The distance between two points and with position vectors and is the magnitude of the displacement vector: .

Geometric properties

Midpoint

The midpoint of the line segment of two points and with position vectors and is .

Parallelograms

Parallelograms have vectors for opposite sides with equal magnitude.

Rhombuses

Rhombuses have vectors for all four sides with equal magnitude.

Differentiation

First principles

Rules

Derivative of

Derivative of , where is a constant:

Derivative of

Increasing and decreasing functions

The sign of the derivative at a point shows whether the function is increasing or decreasing:

If , the function is increasing.
If , the function is decreasing.

Higher derivatives

The second derivative is denoted or . It measures the gradient of the gradient, or the rate of change of the gradient with respect to .

Tangents and normals

For a point on :

The gradient of the tangent is ,
The gradient of the normal is

Stationary points

At the local maxima and minima, . This can be used for optimisation questions.

Nature

Given a stationary point:

If at that point, then it is a local maximum.
If at that point, then it is a local minimum.
If at that point, then no conclusion can be reached.

Integration

Fundamental theorem of calculus

Integration is the opposite of differentiation.

Integral of polynomials

For all rational :

Definite integrals

The definite integral is found by evaluating the integrated expression at the upper limit and subtracting the integrated expression evaluated at the lower limit . This gives the signed area.

If the integrand changes sign between the two limits, then the interval needs to be split to find the total area.