## Section 3.1 Cartesian Coordinates

When we model a relationship between two variables visually, we use the

*Cartesian coordinate system*. This section covers 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 âaddressâ in relation to a starting point. Weâll use a street grid as an analogy. Here is a map with Carlâs 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 (to the right on the map), then go \(3\) blocks north (up on the map).â

Two numbers are used 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 Carlâs 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).

To travel from Carlâs 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 shopâs Cartesian coordinates are \((-3,2)\text{.}\)### Remark 3.1.5.

Itâs 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.### Checkpoint 3.1.6.

Use FigureÂ 2 to answer the following questions.

- What are the coordinates of the bar?
- What are the coordinates of the gas station?
- What are the coordinates of Carlâs house?

### Warning 3.1.7. 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 \((1,3)\) 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 \(3\text{.}\)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 Carlâs neighborhood would look like this:

### Definition 3.1.9. Cartesian Coordinate System.

The Cartesian coordinate system

^{â2â}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 pointâs coordinates, which consists of an \(x\)-coordinate and a \(y\)-coordinate. For example, the point \((1,2)\text{,}\) has \(x\)-coordinate \(1\text{,}\) and \(y\)-coordinate \(2\text{.}\) The origin has coordinates \((0,0)\text{.}\)

A Cartesian coordinate system is divided into four quadrants, as shown in FigureÂ 10. The quadrants are traditionally labeled with Roman numerals.

### Example 3.1.11.

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

## Explanation.

### Reading Questions Reading Questions

#### 1.

What are the coordinates of the gas station in the map of Carlâs neighborhood?

#### 2.

A Cartesian coordinate system has seven âplacesâ within that are worth noting. What are they? (For example, one of them is Quadrant I.)

### Exercises Exercises

#### Identifying 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 \((1,2)\text{.}\)

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

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

#### Creating Sketches of Graphs.

##### 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.

#### Cartesian Plots in Context.

##### 13.

This graph gives the minimum estimates of the wolf population in Washington from 2008 through 2015.

What are the Cartesian coordinates for the point representing the year 2014?

Between 2014 and 2015, 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.

What are the Cartesian coordinates for the point representing the year 1960?

Between 1960 and 1980, 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 \({\left(-7,4\right)}\) is in Quadrant

- I
- II
- III
- IV

.

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

- I
- II
- III
- IV

.

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

- I
- II
- III
- IV

.

The point \({\left(10,2\right)}\) is in Quadrant

- I
- II
- III
- IV

.

##### 16.

The point \({\left(-4,-3\right)}\) is in Quadrant

- I
- II
- III
- IV

.

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

- I
- II
- III
- IV

.

The point \({\left(-9,7\right)}\) is in Quadrant

- I
- II
- III
- IV

.

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

- I
- II
- III
- IV

.

##### 17.

Assume the point \((x,y)\) is 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

.

##### 18.

Assume the point \((x,y)\) is 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

.

##### 19.

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

- Quadrant I
- Quadrant II
- Quadrant III
- Quadrant IV
- the x-axis
- the y-axis

.

#### Plotting Points and Choosing a Scale.

##### 20.

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

##### 21.

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 Cartesian plane with these points marked, and you donât have tick marks that are evenly spaced apart?

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