ORCCA Cartesian Coordinates

Section 3.1 Cartesian Coordinates

Objectives: PCC Course Content and Outcome Guide

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.

Figure 3.1.1. Alternative 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) — Figure 3.1.2. Carl’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 (to the right on the map), then go $3$ blocks north (up on the map).”

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 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) and the path he travels to the restaurant at (2, 3);a line shows his path going east/right 2 blocks and north/up 3 blocks — Figure 3.1.3. Carl’s path to the restaurant

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) and the path he travels to the pet shop (-3,2); a line shows his path going west/left 3 blocks and north/up 2 blocks — Figure 3.1.4. Carl’s path to the pet shop

Remark 3.1.5.

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.

Checkpoint 3.1.6.

Use Figure 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 Carl’s 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 $3\text{.}$

$a Cartesian grid a point marked at (1, 3)$

$a number line with the interval from 1 to 3 shaded$

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 with the points (0,0), (2, 3), (-3,2), (-2,-4), (3, -3), with the horizontal axis labeled x and the vertical axis labeled y — Figure 3.1.8. Carl’s Neighborhood in a Cartesian Coordinate System

Definition 3.1.9. Cartesian Coordinate System.

The Cartesian coordinate system

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, 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 Figure 10. 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. — Figure 3.1.10. A Cartesian grid with four quadrants marked

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.

$a Cartesian grid with arrows drawn to plot each point;(3,2) is right 3, up 2;(-5,-1) is left 5, down 1;(0,-3) is down 3 on the y-axis;(4,0) is right 4 on the x-axis$

$the previous graph with the points drawn and labeled$

Reading Questions Reading Questions

1.

What are the coordinates of the gas station in the map of Carl’s neighborhood?

2.

A Cartesian coordinate system has seven “places” 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 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 $(1,2)\text{.}$

$A=$	$B=$
$C=$	$D=$

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

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

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

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

16.

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

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

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

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

17.

Assume the point $(x,y)$ is in Quadrant II, locate the following points:

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

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

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

18.

Assume the point $(x,y)$ is in Quadrant IV, locate the following points:

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

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

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

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 don’t have tick marks that are evenly spaced apart?

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