## Section4.1Cartesian Coordinates

Â¶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.

###### RenĂ© Descartes

Several conventions used in mathematics are attributed to (or at least named after) RenĂ© Descartes. The Cartesian coordinate system is one of these. You can read about RenĂ© and these conventions at en.wikipedia.org/wiki/RenĂ©_Descartes.

The Cartesian coordinate system identifies the location of every point in a plane. Basically, the system gives every point in a plane its own âaddressâ 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:

âFirst go \(2\) blocks east, then go \(3\) blocks north.â

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).

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

###### 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 FigureÂ 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:

###### Definition4.1.8Cartesian Coordinate System

A 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**. (Visit en.wikipedia.org/wiki/Cartesian_coordinate_system for more.)

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 FigureÂ 4.1.9. The quadrants are traditionally labeled with Roman numerals.

###### 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 pointâs position as an ordered pair, like \((1,2)\text{.}\)

\(A=\) | \(B=\) |

\(C=\) | \(D=\) |

###### 2

Locate each point in the graph:

Write each pointâs 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 2011?

Between 2011 and 2012, the wolf population grew by wolves.

List at least three ordered pairs in the graph.

###### 14

Here is a graph of the foreign-born US population (in millions) during Census years 1960 to 2010. (Source: http://www.pewhispanic.org/2015/09/28/chapter-5-u-s-foreign-born-population-trends/.)

What are the Cartesian coordinates for the point representing the year 1980?

Between 1980 and 2000, the US population that is foreign-born increased by million people.

List at least three ordered pairs in the graph.

Regions in the Cartesian plane.

###### 15

The point \({\left(5,10\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(-7,10\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(-10,-8\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(6,-7\right)}\) is in Quadrant

?

I

II

III

IV

.###### 16

The point \({\left(7,3\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(-9,-5\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(-8,1\right)}\) is in Quadrant

?

I

II

III

IV

.The point \({\left(1,-9\right)}\) is in Quadrant

?

I

II

III

IV

.###### 17

Answer the following questions on the coordinate system:

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

?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

the x-axis

the y-axis

.###### 18

Answer the following questions on the coordinate system:

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 \(y=0\text{,}\) 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 \(x\lt 0 \text{ and } y>0\text{,}\) 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 \(x>0 \text{ and } y\lt 0\text{,}\) 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

.###### 19

Assume the point \((x,y)\) if in Quadrant I, 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 II, 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 donât have tick marks that are evely spaced apart?