X-Y Graphs and the Cartesian Coordinate System
Graphs and solutions to equations - to come.
Polar Coordinate System
Charts and Diagrams (Visual Aids) including Venn Diagrams

X-Y Graphs and the Cartesian Coordinate System

These are also known as rectangular coordinates. You will have come across these graphs quite early in school. The convention is to write the coordinates of a point as (x, y) where x is the number of horizontal steps along the grid, and y is the number of vertical steps up the grid and away from the origin O.

We use equations to understand the relations between variables and constants, The reason we may want to plot an equation on a graph is to visualise those relations. If you are unsure what the plot of an equation looks like, one way to determine it is to identify a set of coordinates that will exist on the plot. In the example below, we first determine a set of coordinates for LaTeX: y=x^2 , where LaTeX: x (the independent variable) goes from LaTeX: -3 to LaTeX: 3 in steps of LaTeX: 1 . We can sketch the plot by connecting the coordinates.

Graphing Example.PNG

Take a look at Desmos or GeoGebra to see what different plots look like.

Resources

Cartesian Coordinates | mathsisfun.com

The Graph of a Function

ASK Coordinate Geometry Overview

More...

This is fun and if you can get 100% you are doing well: Can you identify the 10 equations (some of them standard functions) on this graph? (a timed quiz!) Equations on a Graph Quiz (sporcle.com)

Graphical Solution of Inequalities

Graphs of the Trig Functions

Polar Coordinate System

It sometimes much more convenient to plot angles and distances, instead using the grid of the usual cartesian coordinate system. For example, if we are considering the flow through a circular pipe.

In polar coordinates, we use $LaTeX: \left(\:r,\:\theta\right)$ to represent a point which a distance r from the origin, and where the line from the point to the origin makes an angle of $LaTeX: \theta$ to the horizontal line on the right-hand side of the origin:

How do I graph polar coordinates? | Socratic

Unlike with cartesian axes, the same point can be represented using multiple polar coordinates. For example, the point $LaTeX: \left(4,60^{\circ}\right)$ is the same as $LaTeX: \left(-4,-120^{\circ}\right)$ . Don't believe it? Try it out on Desmos. This is because the following rules apply for polar coordinates:

$LaTeX: \left(r,\theta\right)=\left(r,\theta\pm360^{\circ}\right)=\left(r,\theta\pm2\pi\right)$

$LaTeX: \left(r,\theta\right)=\left(-r,\pm180^{\circ}\right)=\left(-r,\pm\pi\right)$

Conversions between cartesian and polar coordinates can be determined using the following formulae:

Cartesian to Polar	Polar to Cartesian
$LaTeX: r=\sqrt{x^2+y^2}$	$LaTeX: x=\cos\theta$
$LaTeX: \theta=\tan^{-1}\left(\frac{y}{x}\right)$	$LaTeX: y=\sin\theta$

Resources

Polar Coords | desmos.com

First Aid - The Polar Form Worksheet (Worksheet with Answers)

First Aid - Polar Coordinates Worksheet (Worksheet with Answers)

First Aid - Multiplication and Division in Polar Form Worksheet (Worksheet with Answers)

Charts and Diagrams (Visual Aids)

Data and mathematical concepts can be visually represented in various ways, with some being more appropriate than others depending on what's being represented. Here we'll go through some of the most-used types of visual aids you may encounter in engineering, and what situations they may be most appropriate for.

Bar Charts

Bar charts are typically most useful when we are aiming to represent quantities that fall into comparable categories. The most common kinds of bar charts you will encounter are single bar, stacked and clustered bar charts, as shown below.

Bar Chart Examples.PNG

Bars can also be arranged vertically, however when we are dealing with a large number of categories, this horizontal arrangement is more comfortable for the viewer.

Histograms

Histograms are similar to bar charts, but the data falls into intervals as opposed to categories. We call these intervals 'bins', and the size of the interval determines the width of the bar, namely their 'class width'. We may use histograms when we want to observe how data is quantitively distributed.

If the intervals are of equal length, then we can plot a 'uniform histogram', for which the frequency in each bin is determined by the height of the bar. Otherwise, when the intervals are not all of equal length, the height of the bars constitute 'frequency density' and the frequency in each interval is determined by the area of the bar (i.e. $LaTeX: frequency\:density\times class\:width$ ). We can see this in action below:

Histogram Examples-1.PNG

Line and Area Charts

Line and area charts are typically used to demonstrate the change in a quantity (or quantities) with time. These charts can allow us to observe some trends, for example how a business's profits are changing, what average temperatures look like over the course of a year, etc. A line chart can show how individual quantities change with time; an area chart can show how an overall quantity changes as a sum of its constituent parts. Let's see this in action:

Line Chart and Area Chart Example.PNG

Pie Chart

We typically use pie charts to visualise the relative size of a variety of data. Pie charts sometimes get a bad rap, but there are still situations when they can be very useful. It's best to avoid pie charts when we are dealing with a large number of categories.

Pie Chart Example.png

Venn Diagrams

Unlike pie charts, Venn diagrams can help us visualise how data falls into categories, when sometimes those categories overlap. For example, if we are aiming to visualise the subjects that students study in a school, students will of course study more than one subject.

Venn Diagram Example_2.PNG

Venn diagrams also provide us with a useful visual aid for probability problems, which are discussed here.

Scatter Plot

Scatter plots allow us to observe possible correlation between two variables (or three if it's a 3-dimensional plot). Variables can observe positive, negative and no correlation, as shown below.

Correlation Examples.PNG

The strength of the correlation can be measured, and the line of best fit can be calculated. Both of these can be determined using a calculator or Microsoft Excel. See Mathematics - Data (Statistics) for more.

Resources

When Should I Use a Bar Chart? | drawsoft.com

Histograms Overview | BBC Bitesize

The Line Chart - How And When To Use It | infogram.com

When to Use Pie Chart - Things You Should Keep in Mind | edrawsoft.com

Venn diagrams | BBC Bitesize

Venn Diagram Example Questions

A Complete Guide to Scatter Plots | chartio.com

ASK Statistics Overview - Section 1d. Histograms, Section 3a. and 3b. Scatter Plots

Data Visualisation Catalogue | datavizcatalogue.com

Exemplar Infographics | informationisbeautiful.net