Graphing Relations and Networks

Study Notes

Graphs of Relations
- A relation is a set of ordered pairs
- Domain: set of x-values, Range: set of y-values
- Graphing relations on the Cartesian plane
Graphing Techniques
- Table of values method
- Interpolation and extrapolation
- Graphing using symmetry and asymptotes
Graphical Relationships
- Identifying and explaining key features: maxima, minima, asymptotes, intercepts
- Analyzing and interpreting graphs in context

Network Fundamentals
- Nodes (vertices) and arcs (edges)
- Directed and undirected graphs
- Weighted and unweighted networks
Network Representation
- Adjacency matrices and lists
- Drawing and interpreting network diagrams
Shortest Path Algorithms
- Dijkstra's algorithm for single-source shortest paths
- Floyd-Warshall algorithm for all-pairs shortest paths

Annuity Basics
- Defined benefit and defined contribution schemes
- Accumulated value and present value
Annuity Formulae
- Formula for the accumulated value of an annuity: A = P[(1 + r)^n - 1]/r
- Formula for the present value of an annuity: P = A/[(1 + r)^n - 1]/r
Annuity Applications
- Superannuation and retirement savings
- Insurance and investment products

Compound Interest Formulae
- Formula for compound interest: A = P(1 + r/n)^(nt)
- Formula for effective interest rate: r_e = (1 + r/n)^(n) - 1
Compound Interest Concepts
- Effective annual rate and nominal annual rate
- Continuous compounding
Applications of Compound Interest
- Investment and loan calculations
- Comparing investment options

A relation is a set of ordered pairs, with a domain (set of x-values) and a range (set of y-values).
Graphing relations on the Cartesian plane involves plotting points and identifying key features.
The table of values method is a technique for graphing relations by creating a table of x and y values.
Interpolation and extrapolation are used to estimate values between or outside the given data points.
Graphing using symmetry and asymptotes involves identifying and utilizing these properties to graph relations.

A network consists of nodes (vertices) connected by arcs (edges), which can be directed or undirected.
Weighted networks have numerical values assigned to each arc, while unweighted networks do not.
Adjacency matrices and lists are used to represent networks, with adjacency matrices displaying the connections between nodes.
Drawing and interpreting network diagrams involves understanding the relationships between nodes and arcs.
Dijkstra's algorithm is used to find the shortest path in a network from a single source, while Floyd-Warshall algorithm finds the shortest path between all pairs of nodes.

An annuity is a series of fixed payments made at regular intervals, with a defined benefit or defined contribution scheme.
The accumulated value of an annuity is the total value of the payments at a given time, while the present value is the current value of future payments.
The formula for the accumulated value of an annuity is A = P[(1 + r)^n - 1]/r.
The formula for the present value of an annuity is P = A/[(1 + r)^n - 1]/r.
Annuities are used in superannuation and retirement savings, as well as insurance and investment products.

Compound interest involves earning interest on both the principal and accrued interest, resulting in exponential growth.
The formula for compound interest is A = P(1 + r/n)^(nt), where A is the accumulated value, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.
The formula for effective interest rate is r_e = (1 + r/n)^(n) - 1, which is the rate of interest earned per year.
The effective annual rate is the rate of interest earned per year, while the nominal annual rate is the rate of interest before compounding.
Continuous compounding involves compounding interest continuously, resulting in the highest possible rate of return.
Compound interest is used in investment and loan calculations, and is essential for comparing investment options.