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Section4.1Cartesian Coordinates

When we model relationships with graphs, we use the Cartesian coordinate system. This section covers the basic vocabulary and ideas that come with the Cartesian coordinate system.

Figure4.1.1Alternative Video Lesson

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.

a Cartesian grid with Carl's home at (0,0), a restaurant at (2, 3), a pet shop at (-3,2), a gas station at (-2,-4), and a bar at (3, -3)
Figure4.1.2Carl's neighborhood

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, then go \(3\) blocks north.”

Carl uses two numbers to locate the restaurant. In the Cartesian coordinate system, these numbers are called coordinates and they are written as the ordered pair \((2,3)\text{.}\) The first coordinate, \(2\text{,}\) represents distance traveled from Carl's house to the east (or to the right horizontally on the graph). The second coordinate, \(3\text{,}\) represents distance to the north (up vertically on the graph).

a Cartesian grid with Carl's home at (0,0), a restaurant at (2, 3), a pet shop at (-3,2), a gas station at (-2,-4), and a bar at (3, -3)
Figure4.1.3Carl's neighborhood

Alternatively, 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 \((-3,2)\text{.}\)

a Cartesian grid with Carl's home at (0,0), a restaurant at (2, 3), a pet shop at (-3,2), a gas station at (-2,-4), and a bar at (3, -3)
Figure4.1.4Carl's neighborhood
Remark4.1.5

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.

Notation Issue: Coordinates or Interval?

Unfortunately, the notation for an ordered pair looks exactly like interval notation for an open interval. Context will help you understand if \((2,3)\) indicates the point \(2\) units right of the origin and \(3\) units up, or if \((2,3)\) indicates the interval of all real numbers between \(2\) and \(3\text{.}\)

Exercise4.1.6

Use Figure 4.1.2 to answer the following questions.

Traditionally, the variable \(x\) represents numbers on the horizontal axis, so it is called the \(x\)-axis. The variable \(y\) represents numbers on the vertical axis, so it is called the \(y\)-axis. The axes meet at the point \((0,0)\text{,}\) which is called the origin. Every point in the plane is represented by an ordered pair, \((x,y)\text{.}\)

In a Cartesian coordinate system, the map of Carl's neighborhood would look like this:

a Cartesian grid points \((0,0), (2, 3), (-3,2), (-2,-4),\) and \((3, -3)\text{,}\) with \(x\)-axis and \(y\)-axis labeled
Figure4.1.7Carl's Neighborhood in a Cartesian Coordinate System
Definition4.1.8Cartesian Coordinate System

A Cartesian coordinate system 2 en.wikipedia.org/wiki/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, measured in the same unit of length. 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 \(x\)-axis and \(y\)-axis.

The plane based on the \(x\)-axis and \(y\)-axis is called a coordinate plane. The ordered pair used to locate a point is called the point's coordinates, which consists of an \(x\)-coordinate and a \(y\)-coordinate. For example, for the point \((1,2)\text{,}\) its \(x\)-coordinate is \(1\text{,}\) and its \(y\)-coordinate is \(2\text{.}\) The origin has coordinates \((0,0)\text{.}\)

A Cartesian coordinate system is divided into four quadrants, as shown in Figure 4.1.9. The quadrants are traditionally labeled with Roman numerals.

a Cartesian grid with Quadrant I marked in the top right section, Quadrant II marked in the top left section, Quadrant III marked in the bottom left section, Quadrant IV marked in the bottom right section.
Figure4.1.9A Cartesian grid with four quadrants marked
Example4.1.10

On paper, sketch a Cartesian coordinate system with units, and then plot the following points: \((3,2),(-5,-1),(0,-3),(4,0)\text{.}\)

Subsection4.1.1Exercises

Identify coordinates.

1

Locate each point in the graph:

Write each point’s position as an ordered pair, like \((1,2)\text{.}\)

\(A=\) \(B=\)
\(C=\) \(D=\)
2

Locate each point in the graph:

Write each point’s position as an ordered pair, like \((1,2)\text{.}\)

\(A=\) \(B=\)
\(C=\) \(D=\)

Make some sketches.

3

Sketch the points \((8,2)\text{,}\) \((5,5)\text{,}\) \((-3,0)\text{,}\) and \((2,-6)\) on a Cartesian plane.

4

Sketch the points \((1,-4)\text{,}\) \((-3,5)\text{,}\) \((0,4)\text{,}\) and \((-2,-6)\) on a Cartesian plane.

5

Sketch the points \((208,-50)\text{,}\) \((97,112)\text{,}\) \((-29,103)\text{,}\) and \((-80,-172)\) on a Cartesian plane.

6

Sketch the points \((110,38)\text{,}\) \((-205,52)\text{,}\) \((-52,125)\text{,}\) and \((-172,-80)\) on a Cartesian plane.

7

Sketch the points \((5.5,2.7)\text{,}\) \((-7.3,2.75)\text{,}\) \(\left(-\frac{10}{3},\frac{1}{2}\right)\text{,}\) and \(\left(-\frac{28}{5},-\frac{29}{4}\right)\) on a Cartesian plane.

8

Sketch the points \((1.9,-3.3)\text{,}\) \((-5.2,-8.11)\text{,}\) \(\left(\frac{7}{11},\frac{15}{2}\right)\text{,}\) and \(\left(-\frac{16}{3},\frac{19}{5}\right)\) on a Cartesian plane.

9

Sketch a Cartesian plane and shade the quadrants where the \(x\)-coordinate is negative.

10

Sketch a Cartesian plane and shade the quadrants where the \(y\)-coordinate is positive.

11

Sketch a Cartesian plane and shade the quadrants where the \(x\)-coordinate has the same sign as the \(y\)-coordinate.

12

Sketch a Cartesian plane and shade the quadrants where the \(x\)-coordinate and the \(y\)-coordinate have opposite signs.

These exercises have Cartesian plots with some context.

13

This graph gives the minimum estimates of the wolf population in Washington from 2008 through 2015. (Source: http://wdfw.wa.gov/publications/01793/wdfw01793.pdf)

What are the Cartesian coordinates for the point representing the year 2008?

Between 2008 and 2009, the wolf population grew by wolves.

List at least three ordered pairs in the graph.

Regions in the Cartesian plane.

15

The point \({\left(-5,-6\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(-6,2\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(5,-2\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(2,2\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

16

The point \({\left(-2,8\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(3,-3\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(1,1\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \({\left(-10,-9\right)}\) is in Quadrant

  • I

  • II

  • III

  • IV

.

17

Answer the following questions on the coordinate system:

For the point \((x,y)\text{,}\) if \(x\lt 0 \text{ and } y\lt 0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x>0 \text{ and } y\lt 0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x>0 \text{ and } y>0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x\lt 0 \text{ and } y>0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(y=0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x=0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

18

Answer the following questions on the coordinate system:

For the point \((x,y)\text{,}\) if \(x>0 \text{ and } y\lt 0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x\lt 0 \text{ and } y>0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x\lt 0 \text{ and } y\lt 0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x=0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(y=0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

For the point \((x,y)\text{,}\) if \(x>0 \text{ and } y>0\text{,}\) then the point is in/on

  • Quadrant I

  • Quadrant II

  • Quadrant III

  • Quadrant IV

  • the x-axis

  • the y-axis

.

19

Assume the point \((x,y)\) if in Quadrant IV, locate the following points:

The point \((-x,y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \((x,-y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \((-x,-y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

20

Assume the point \((x,y)\) if in Quadrant IV, locate the following points:

The point \((-x,y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \((x,-y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

The point \((-x,-y)\) is in Quadrant

  • I

  • II

  • III

  • IV

.

Writing questions.

21

What would be the difficulty with trying to plot \((12,4)\text{,}\) \((13,5)\text{,}\) and \((310,208)\) all on the same graph?

22

The points \((3,5)\text{,}\) \((5,6)\text{,}\) \((7,7)\text{,}\) and \((9,8)\) all lie on a straight line. What can go wrong if you make a plot of a Cartesion plane with these points marked, and you don’t have tick marks that are evely spaced apart?