Graphing Relations and Networks

Graphing and Relations

• 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

Networks and Decision Maths

• 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

Annuities

• 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

• 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

Graphing and Relations

• 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.

Networks and Decision Maths

• 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.

Annuities

• 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

• 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.