Maths Methods Unit 1 / Mathematical Methods Unit 1

Mathematical methods unit 1

Define set

A collection of elements, usually a group of numbers

Define element

the members of a set; a ∈ A means a is an element of, or belongs to, the set A. If a is not an element of the set A, this is written as a ∉ A

State what {...} represents

Refers to a set of something

State what ∈ represents

Is an element of

State what ∉ represents

Is not an element of

State what ⊂ represents

Is a subset of

State what ⊂/ represents

Is not a subset of

State what ∩ represents

Intersection with

State what ∪ represents

Union with

State what \ means

Excluding

State what ∅ means

A null set or empty set

State what {a, b, c} represents

Is a set of three letters

State what {(a, b), (c, d),...}{(a, b), (c, d),...} represents

Is an infinite set of ordered pairs

Define relation

A set of ordered pairs

State what N represents

The set of natural numbers

State what J represents

The set of integers

State what Q represents

The set of rational numbers

State what R represents

The set of Real numbers

State what R+ refers to

The set of positive Real numbers

State what R- refers to

The set of negative Real numbers

Define interval

A set of numbers described by two numbers where any number that lies between those two numbers is also included in the set

Describe interval notation

It is a convenient way to represent an interval using only the end values and indicating whether those end values are included or excluded.

State what a rounded bracket represents in interval notation

A rounded bracket indicates a value that is excluded.

State what a square bracket represents in interval notation

A square bracket indicates a value that is included.

Describe what an open circle represents on a number line or Cartesian plane

An open circle represents an excluded value

Describe what a closed circle represents on a number line or Cartesian plane

A closed circles represents an included value

Describe a mathematical relation

It is any set of ordered pairs. The ordered pairs may be listed, described by a rule, such as an equation or inequation, or presented as a graph.

Define continuous

All values within a specified interval are permitted

Define discrete

Only fixed values are permitted

Define independent variable

The first value, or x-value, in a set of ordered pairs

Define dependent variable

The second value, or y-value, in a set of ordered pairs

Define a one-to-one relationship

A one-to-one relation exists if for any x-value there is only one corresponding y-value and vice versa. This is a function. (Linear)

Define a one-to-many relationship

A one-to-many relation exists if for any x-value there is more than one y-value, but for any y-value there is only one x-value. This is not a fucntion. (Hyperbola)

Define a many-to-one relationship

A many-to-one relation exists if there is more than one x-value for any y-value, but for any x-value there is only one y-value. This is a function. (Parabola)

Define a many-to-many relationship

A many-to-many relation exists if there is more than one x-value for any y-value and vice versa. This is not a function. (Circles)

Define functions

A relation in which ordered pairs of unique numbers can be identified by a given rule or equation

Describe the vertical line test

A test used to identify a function and can be applied by placing a vertical line (parallel to the y-axis) through the graph. If there is one intersection, the graph is a function. If the line intersects the graph more than once, while remaining parallel to the y-axis, then the graph does not represent a function. All polynomial relations are functions.

Define domain

The set of all x-values of the ordered pairs (x,y) that make up a relation

Define range

The set of all y-values of the ordered pairs (x,y) that make up a relation

Describe how to determine the domain and range from a given rule

If a relation is described by a rule, it should also specify the domain. For example: the relation {(x, y):y=2x,x∈{1, 2, 3}} describes the set of ordered pairs {(1, 2), (2, 4), (3, 6)}.

The domain is the set X={1, 2, 3}X=1, 2, 3, which is given.

The range is the set Y={2, 4, 6}Y=2, 4, 6, and can be found by applying the rule y = 2xy = 2x to the domain values.

Define implied domain

The set of x-values for which a rule has meaning.

For example:

{(x, y):y = x3} has the implied domain R.

{(x,y):y = √x} has the implied domain x≥0, where x∈R, since the square root of a negative number is an imaginary value.

Describe restricted domain

For some practical situations, restrictions have been placed on the values of the variables in some polynomial models. These usually effect the range.

Define restricted domain

A subset of a function's maximal domain, often due to practical limitations on the independent variable in modelling situations

Describe function notation

The rule for a function such as y=x^2 will often be written as f(x)=x^2. This is read as 'f of x equals x^2'.

Functions are referred to as y=f(x), particularly when graphing a function as the set of ordered pairs (x, y).

Define image

A figure after a transformation; for the function x→f, f(x) is the image of x under the mapping f

Define mapping

A function that pairs each element of a given set (the domain) with one or more elements of a second set (the range)

Define codomain

The set of all y-values available for pairing with x-values to form a mapping according to a function rule y=f(x). Not necessarily every element of the codomain will be within the range.

Describe formal function notation for x -> x^2

f: R -> R, f(x) = x^2

name of function: domain of f -> codomain, rule for equation of f

Define transformations

Geometric operators that may change the shape and/or position of a graph

Define dilation

A linear transformation that enlarges or reduces the size of a figure by a scale factor k parallel to either axis or both axes (stretching or compressing)

Define reflections

Transformations of a figure defined by the line of reflection where the image point is a mirror image of the pre-image point (flipping)

Define translations

Transformations of a figure where each point in the plane is moved a given distance in a horizontal or vertical direction (moving horizontally or vertically)

Describe dilation of a function from the x-axis

A dilation from the x-axis acts parallel to the y-axis.

The point (x, y)→(x, ay) when dilated by a factor a from the x-axis.

A dilation of factor a from the x-axis transforms y=x^2 to y=ax^2 and, generalising, under a dilation of factor a from the x-axis, y=f(x)→y=af(x)

State the general rule for dilating a function from the x-axis

y=af(x) is the image of y=f(x) under a dilation of factor a from the x-axis, parallel to the y-axis.

Describe dilation of a function from the y-axis

A dilation from the y-axis acts parallel to the x-axis, or in the x-direction.

The point (x, y)→(bx, y) when dilated by a factor b from the y-axis.

Dilating y=f(x) by a factor 2 from the y-axis gives the image y=f(x/2) for example.

State the general rule for dilating a function from the y-axis

y=f(bx) is the image of y=f(x) under a dilation of factor 1/b from the y-axis, parallel to the x-axis.

State the general rule for reflecting a function in the y-axis

y=f(−x) is the image of y=f(x) under a reflection in the y-axis

State the general rule for reflecting a function in the x-axis

y=−f(x) is the image of y=f(x) under a reflection in the x-axis

State the general rule for translating a function horizontally

y=f(x+c) is the image of y=f(x) under a horizontal translation of c units to the left.

A negative c-value will result in translation to the right.

When C is positive -> move left

When C is negative -> move right

State the general rule for translating a function vertically

y=f(x)+d is the image of y=f(x) under a vertical translation of d units upwards.

A negative d-value will result in translation downwards.

When D is positive -> move up

When D is negative -> move down

Describe the graph of y=af(b(x+c))+d

• a gives the dilation factor |a||a| from the x-axis, parallel to the y-axis.

• If a<0, there is a reflection in the x-axis.

• b gives the dilation factor 1/b from the y-axis

• If b<0, there is a reflection in the y-axis.

• c gives the horizontal translation parallel to the x-axis.

• d gives the vertical translation parallel to the y-axis.

State the order for transforming functions

Any dilation or reflection should be applied before any translation.

Define Piece-wise function

A function whose rule takes different forms for different sections of its domain. If branches of a piece-wise function joins, it is continuous. If it does not, it is discontinuous for that part of the domain.

Define modelling

Uses mathematical concepts to describe the behaviour of a system, usually in the form of equations