- What are the coordinates of the bar? ,
- What are the coordinates of the gas station? ,
- What are the coordinates of Carl’s house? ,
We plot points on a 2-dimensional Cartesian graph.
Graphing 2-D Cartesian Coordinates
Metacognitive Moment The purpose of math is to make sense of the world
around up and phenomena we encounter. We use analogies to ground our learning in
familiar situations to help us understand mathematics through connections between
math and the world.
When we model a relationship between two variables visually, we often use the Cartesian coordinate system. We begin with the basic vocabulary and ideas that come with the Cartesian coordinate system.
The Cartesian coordinate system identifies the location of every point in a plane. Basically, the system gives every point in a plane its own “address” in relation to a starting point. We’ll use a street grid as an analogy. Here is a map with Carl’s home at the center. The map also shows some nearby businesses. Assume each unit in the grid represents one city block.
If Carl has an out-of-town guest who asks him how to get to the restaurant, Carl could say, “First go 2 blocks east (to the right on the map), then go 3 blocks north (up on the map).” Altnernatively, we could say, “First go 3 blocks north (up on the map), then go 2 blocks east (to the right on the map).” (There are actually many different paths, especially if we consider the blocks as open fields the friend can walk through.)
Two numbers are used to locate the restaurant. In the Cartesian coordinate system, these two numbers together are called coordinates and they are written as an ordered pair. The ordered pair (2,3) represents the location of the restaurant. The first coordinate, 2, represents distance traveled from Carl’s house to the east (or to the right horizontally on the graph). The second coordinate, 3, represents distance to the north (up vertically on the graph).
To travel from Carl’s home to the pet shop, he would go 3 blocks west, and then 2 blocks north.
In the Cartesian coordinate system, the positive directions are to the right horizontally and up vertically. The negative directions are to the left horizontally and down vertically. So the pet shop’s Cartesian coordinates are .
It’s important to know that the order of Cartesian coordinates is (horizontal,
vertical). This idea of communicating horizontal information before vertical
information is consistent throughout most of mathematics.
Use the graph of Carl’s neighborhood above to answer the following questions.
Notation Issue Unfortunately, the notation for an ordered pair looks exactly like
interval notation for an open interval. Context will help you understand if (1,3)
indicates the point 1 unit right of the origin and 3 units up,
or if (1,3) indicates the interval of all real numbers between 1 and 3.
Traditionally, the variable represents numbers on the horizontal axis, so it is called the -axis. The variable represents numbers on the vertical axis, so it is called the -axis. The axes meet at the point , which is called the origin. Every point in the plane is represented by an ordered pair, .
In a Cartesian coordinate system, the map of Carl’s neighborhood would look like this:
The Cartesian coordinate system is a coordinate system that specifies each point
uniquely in a plane by a pair of numerical coordinates, which are the signed
(positive/negative) distances to the point from two fixed perpendicular directed lines.
Those two reference lines are called the horizontal axis and vertical axis, and the
point where they meet is the origin. The horizontal and vertical axes are often called
the -axis and -axis.
The plane based on the -axis and -axis is called a coordinate plane. The ordered
pair used to locate a point is called the point’s coordinates, which consists of an
-coordinate and a -coordinate. For example, the point , has -coordinate 1, and
-coordinate 2. The origin has coordinates .
A Cartesian coordinate system is divided into four quadrants that are separated by the axes, as shown below. The quadrants are traditionally labeled with Roman numerals.
