Skip to main content

Section 1 Coordinate Plane Overview

In this module we will be talking a look at lines. The main learning objective here is to understand some of the notation surrounding lines, be able to plot them, and finally, work with word problems involving lines. The content here is intended as an introduction to lines, but works well as a refresher if you are getting started with a subject that has lines at the focus (linear regression). To get started, we are going to explore the coordinate plane. This will be where we will plot lines (and many other function and relations later).

Subsection 1.1 What is the Coordinate Plane?

The coordinate plane can be thought of as a reference space where we can display related quantities. For example, if we have a way to determine population given time, then we can use the coordinate plane as a way to showcase the connections between these two quantities. Let's take a look at the overall structure of the coordinate plane.

Figure 1.1. The coordinate plane.

Each of the four quadrants are traditionally indicated with a roman numeral. The "first quadrant" is the upper left, and then we count to four counterclockwise. This is the main verbal point of reference for where we are at, so it is worth committing this to memory.

The point at the center where the axes cross is called the origin. The horizontal axis is often called the x-axis, and the vertical axis the y-axis. It should be noted that this is not set in stone and can be modified for our applications.

We will navigate the coordinate plane by starting at the origin and moving left and right, then up and down. In the next subsection we will look at this more, but this will be the base idea behind graphing.

As mentioned above, we can relate quantities visually using this tool. We will explore this more later, but for now let's take a look at what that would look like.

Figure 1.2. The coordinate plane.

The goal of a graph like this is to give the reader an overview of the connection between time and population. So when we start to read and work with graphs, looking at the axes and establishing a understanding of the units and quantities is a great first step.

The above graph is getting a bit ahead of ourselves, so to start, let's talk a closer look at plotting points.

Subsection 1.2 Plotting Points

In this section we are going to take a look at how we will plot points and what "moving around" in the coordinate plane will look like and mean.

Figure 1.3. The coordinate plane.

Right is generally going to refer to moving in the positive x direction (left in the negative x direction). Up is generally going to refer to moving in the positive y directions (down in the negative y direction). Note that if either axis is labeled differently, the reference changed a bit, but the language will stay consistent.

Movement in the Coordinate Plane.

Now the idea behind capturing the movement is going to be collected in a mathematical point. The notation will (generally) be,

\begin{equation*} (\textrm{left/right},\textrm{up/down}) \end{equation*}

So the point \((4,5) \) would mean, right 4 and up 5.

Another way to look at it is that the first value in the point refers to the horizontal movement, and the second value refers to the vertical movement. The language that we will start with as we work with points is, "plot the following points." This means, place a small reference dot at the location indicated by the given point. Often in questions we will be asked to plot a few points. Usually we will do this all on the same coordinate plane. Let's take a look at an example.

Draw a blank coordinate plane and label the horizontal axis, "x-axis" and the vertical axis, "y-axis". Then make 10 equal distant marks on each axis and label them from \(-5\) to \(5 \) (click the solution to get a sneak peek).

On the coordinate plane, plot the each of the five points,

  1. (0,0)
  2. (1,1)
  3. (-3,-3)
  4. (0,-4)
  5. (-1,3)

Label each of them with the \((x,y) \) point close to the dot.

Solution
Figure 1.5. The coordinate plane.

The previous example showcased how we will orient ourselves in the coordinate plane. With this in mind, we can add the additional context of the other example that we have been considering. This next example will allow us to explore plotting points in the coordinate plane with time as the horizontal axis and population as the vertical axis.

Suppose that we are running an experiment studying the impact of dish-soap on the population of bacteria. At 1 minute, the population of bacteria is 2 colony-forming units (Cfu/ml). After 4 more minutes, the population of bacteria is 1 Cfu/ml.

The above observation has two different points that can be plotted. Plot both of these points as we did in the previous example and label them, "Check 1" and "Check 2".

Solution
Figure 1.7. Plotting two points (time v population).

We can see that the population has decreased, and the graph gives us a visual description of this decrease.

Section Conclusion.

So as we close this section, let's reflect on what we have explored:
  1. The coordinate plane has two axes (Pronounced ak-seez), that we often label \(x \) and \(y \text{.}\)
  2. We also have labels for the quadrants. Starting in the upper right, that is quadrant I, then moving counterclockwise, we have II, III, and IV.
  3. We can capture information in the plane by moving left or right and then up or down, and logging this as a dot in the plane that plots the point.
  4. The traditional order for a point is (horizontal, vertical).

Moving forward, we are going to explore the questions, What if we want to plot many points? Is there a way that we can encode a collection of relationships in one piece of mathematical notation?