Chapters

## What is a Plane?

**coordinates**are and how to plot them, we should first understand what a plane is. When we talk about a plane in math, we’re usually referring to a Cartesian plane. Take a look at the properties of the

**Cartesian**plane below.

Property | Example | ||

A | Two perpendicular lines | Two lines at 90 degrees from each other | X-axis and y axis |

B | Axes | Axes that have numbers on them | Think of two number lines intersecting each other |

C | X-axis | Holds x coordinates | For point (3,4), the x coordinate is 3 |

D | Y-axis | Holds y coordinates | For point (3,4), the y coordinate is 4 |

The Cartesian plane is named after Rene **Descartes**, a French mathematician. The Cartesian plane is simply referred to as an axis, graph or plot.

## Coordinates

The reason we have a graph is to be able to visualize coordinates. You may have heard the word coordinates to describe the **physical location** of something on earth. For example, global positioning systems, or GPS, use coordinates to locate various things.

Coordinates in math follow the same idea. Let’s start with a simple example: coordinates on a** real number line**. This is also referred to as a one dimensional coordinate system.

As you can see, a number line can, a number line can include any range of real numbers. A coordinate on a number line is simply one number. Let’s plot **three** numbers on a number line:

A | 1 |

B | 5 |

C | -3 |

When we talk about coordinates on a Cartesian plane, we are referring to a **two-dimensional** coordinate system. This means we have two points to graph: an x and a y coordinate.

The x coordinate is always on the left while the y coordinate is always on the right. Let’s take a few examples and **graph** them:

A | (4,5) |

B | (-3,2) |

C | (0,0) |

## Quadrants

Another important property of a graph is that it has **4 quadrants**. Take a look below for a definition of what these quadrants mean:

x | y | |

Quadrant 1 | 4 | 5 |

Quadrant 2 | -3 | 2 |

Quadrant 3 | -4 | -3 |

Quadrant 4 | 2 | -4 |

The table above gives us the value for each point in the quadrant in the graph. Do you notice any **pattern** here? Depending on the quadrant, the x and y coordinates will always be either positive or negative. Take a look at the table below.

x | y | (x,y) | Example | |

Quadrant 1 | + | + | (+,+) | (4,5) |

Quadrant 2 | - | + | (-,+) | (-3,2) |

Quadrant 3 | - | - | (-,-) | (-4,-3) |

Quadrant 4 | + | - | (+,-) | (2,-4) |

## Origin

The origin of a graph is located at the very **centre** of the plane. Let’s take a look at the origin on the graph.

As you can see, the coordinates for the **origin** are always the same.

The origin is very useful when we want to** plot** coordinates. It tells us where to start in order to begin moving up or down on the graph. Take a look at the image below for an example.

## Horizontal Axis

The x axis is also called the horizontal axis. This is because it lies horizontally on the Cartesian plane. The **horizontal axis** touches all four quadrants, however it only goes in two directions. Let’s take a look at two examples.

A | (3,4) | Right 3 |

B | (-2,1) | Left 2 |

C | (-3,-3) | Left 3 |

The x coordinate tells us **how many** places to the left or to the right we have to go.

## Vertical Axis

The y axis is also called the vertical axis. This is because it lies **vertically** on the Cartesian plane. The vertical axis touches all four quadrants, however it only goes in two directions. Let’s take a look at two examples.

A | (3,4) | Up 4 |

B | (-2,1) | Up 1 |

C | (-5,-4) | Down 4 |

The y coordinate tells us how many places **up** or **down** we have to go.

## Constant

While there are many different types of functions we can graph on a plane, the most common one is a linear function. A **linear function** involves any straight line relationship, whose standard form is written as follows:

If we were to plug in values for the x variable, we get a corresponding y variable. A **line** is essentially a set of coordinates that can be connected together.

x | y | |

A | 1 | 7 |

B | 2 | 10 |

C | 3 | 13 |

D | 4 | 16 |

E | 5 | 19 |

Sometimes, however, we have a linear equation that only has two elements: the y variable and a **constant** term.

This graph looks like just a **straight line**. Let’s take a look at a few examples.

A | B |

y = 3 | y = -2 |

## Example 1

Let’s test our knowledge on quadrants. Looking at the quadrant specified below, can you give a **list** of coordinates that would belong to this quadrant?

Here are some possible answers:

- (-5,3)
- (-2, 10)
- (-7,8)

## Example 2

**list**of coordinates below, can you tell which quadrant they belong to?

- (-5, -5)
- (-2, -1)
- (-6, -3)

Because both the x and y values have a **negative value**, we are looking at the 3rd quadrant.

## Example 3

Constants can sometimes be tricky. Graph the following **constants**:

A | B | C |

y = 2 | x = 3 | y = -5 |

Because x and y are constants here, we draw horizontal and vertical **lines**.

## Example 4

Let’s plot the coordinates to the following **equation**.

Here is the graph:

The platform that connects tutors and students