Patterns+and+Graphing

=Try this coordinate [|game] =

Practice test answers: = = =Cartesian Coordinates = Cartesian coordinates can be used to pinpoint where you are on a map or graph.

Cartesian Coordinates
Using Cartesian Coordinates you mark a [|point] on a graph by **how far along** and **how far up** it is:
 * [[image:http://www.mathsisfun.com/images/coordinates-cartesian.gif width="340" height="230" caption="graph with point (12,5)"]] ||
 * The point **(12,5)** is 12 units along, and 5 units up. ||

X and Y Axis

 * [[image:http://www.mathsisfun.com/data/images/x-axis.gif width="94" height="53" caption="x axis"]] || The //left-right// (**horizontal**) direction is commonly called **X**. ||
 * [[image:http://www.mathsisfun.com/data/images/y-axis.gif width="49" height="70" caption="y axis"]] || The //up-down// (**vertical**) direction is commonly called **Y**. ||

... and you are ready to go ||
 * [[image:http://www.mathsisfun.com/data/images/x-y-axes.gif width="251" height="176"]] || Put them together on a graph ...
 * Where they cross over is the "0" point,

 The plural of Axis is **//Axes//**, and is pronounced //ax-eez// 
 * you measure everything from there**.
 * The **X Axis** runs horizontally through zero
 * The **Y Axis** runs vertically through zero ||
 * Axis**: The reference line from which distances are measured.

Example:
Point **(6,4)** is 6 units across (in the **x** direction), and 4 units up (in the **y** direction)

So **(6,4)** means: Go along 6 and then go up 4 then "plot the dot".

And you can remember which axis is which by: x is A CROSS, so x is ACROSS the page. Now would be a good time to play with [|Interactive Cartesian Coordinates] to see for yourself how it all works. ||
 * [[image:http://www.mathsisfun.com/data/images/interactive-cartesian-coordinates.gif width="150" height="150" caption="interactive-cartesian-coordinates" link="http://www.mathsisfun.com/data/cartesian-coordinates-interactive.html"]] ||  || ==Play With It ! ==

Like 2 Number Lines Put Together
It is like we put two [|Number Lines] together, one going left-right, and the other going up-down.

Direction

 * [[image:http://www.mathsisfun.com/images/style/right-arrow.gif width="46" height="46"]] || As **x** increases, the point moves further **right**.

When x decreases, the point moves further to the left. ||
 * [[image:http://www.mathsisfun.com/images/style/up.gif width="46" height="46"]] || As **y** increases, the point moves further **up**.

When y decreases, the point moves further down. ||

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Writing Coordinates
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">The coordinates are always written in a certain order: <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">This is called an "**ordered pair**" (a **pair** of numbers in a special **order**) <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this: (3,2) <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> In other words, only 5 units up.
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">the horizontal distance first,
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">then the vertical distance.
 * Example: (3,2) means 3 units to the right, and 2 units up**
 * Example: (0,5) means 0 units to the right, and 5 units up.**

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">The Origin
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">The point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Abscissa and Ordinate
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">You may hear the words "Abscissa" and "Ordinate" ... they are just the x and y values:
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Abscissa: the horizontal ("x") value in a pair of coordinates: how far **along** the point is
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Ordinate: the vertical ("y") value in a pair of coordinates: how far **up or down** the point is

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">"Cartesian" ... ?
<span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> They are called //Cartesian// because the idea was developed by the mathematician and philosopher **Rene Descartes** who was also known as**//Cartesius//**. He is also famous for saying //"I think, therefore I am"//.

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">What About Negative Values of X and Y?
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Just like with the Number Line, you can also have negative values. Negative: start at zero and **head in the opposite direction**: <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Negative x goes **to the left**
 * <span style="color: #000088; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Negative y goes **down**
 * [[image:http://www.mathsisfun.com/data/images/coords-4-quadrants.gif width="313" height="233"]] || So, for a negative number:
 * go backwards for x
 * go down for y

For example **(-6,4)** means:

go **back** along the x axis 6 then go up 4.

And **(-6,-4)** means:

go **back** along the x axis 6 then go **down 4.** || When we include negative values, the x and y axes divide the space up into 4 pieces: **Quadrants I, II, III** and **IV** //(They are numbered in a <span style="background-color: rgba(180,230,255,0.2);">[|counterclockwise] direction)// || || <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> In **Quadrant I** both x and y are positive, but ... <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Like this: <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> (horizontal) ||~ Y (vertical) ||~ Example || <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> Example: The point "A" (3,2) is 3 units along, and 2 units up. Both x and y are positive, so that point is in "Quadrant I" <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction). Both x and y are negative, so that point is in "Quadrant III" <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Note: The word **Quadrant** comes form //quad// meaning **four**. For example, four babies born at one birth are called //quadruplets//, a four-legged animal is a//quadruped//. and a //quadrilateral// is a four-sided polygon.
 * || ==<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Four Quadrants ==
 * in **Quadrant II** x is negative (y is still positive),
 * in **Quadrant III** both x and y are negative, and
 * in **Quadrant IV** x is positive again, while y is negative.
 * [[image:http://www.mathsisfun.com/data/images/cartesian-coordinates.gif width="284" height="286"]] ||  ||~ Quadrant ||~ X
 * **I** || Positive || Positive || (3,2) ||
 * **II** || //Negative// || Positive ||  ||
 * **III** || //Negative// || //Negative// || (-2,-1) ||
 * **IV** || Positive || //Negative// ||  ||   ||

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">Dimensions: 1, 2, 3 and more ...
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Think about this: <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;"> so any position needs just **one number** || so any position needs **two numbers** || that is **three numbers**, or 3 dimensions! ||
 * ~ 1 || The <span style="background-color: rgba(180,230,255,0.2);">[|Number Line] can only go:
 * left-right
 * ~ 2 || Cartesian coordinates can go:
 * left-right, and
 * up-down
 * ~ 3 || How do we locate a spot in the real world (such as the tip of your nose)? We need to know:
 * left-right,
 * up-down, and
 * forward-backward,

<span style="color: #993300; font-family: qarmic,Verdana,sans-serif; font-size: 20px;">3 Dimensions
<span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">Cartesian coordinates can be used for locating points in 3 dimensions as in this example: three-dimensional Cartesian coordinates. || <span style="font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 15px;">In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you! <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> <span style="display: block; font-family: Verdana,Arial,Tahoma,sans-serif; font-size: 17px;"> [|Interactive Cartesian Coordinates][|Hit the Coordinate Game][|Polar and Cartesian Coordinates][|The Number Line][|Graphs Index]
 * [[image:http://www.mathsisfun.com/data/images/cartesian-3d.gif width="316" height="240"]] ||
 * Here the point **(2, 4, 5)** is shown in