Sunday, July 14, 2013

# What is an XY coordinate plane? GRE Math fact

2:41 AM
GRE Quant questions involving coordinate geometry and XY plane are very common. Do you understand basics of a coordinate plane? Let’s have a quick refresher:
Remember number line?
If you recall, a number line is a representation of real numbers, both positive and negative, and integers and decimals on a line that extends to infinity in both directions.  Numbers are represented on the number line in the form of equal partitions marked on a horizontal line, as shown below.   Although this image only shows the integers from -6 to 6, the line includes all real numbers, continuing towards infinity in each direction.

Well, you get it...don’t you? Question on your mind is "why are we telling you about number line?" You see, just like a number line represents a number in one dimension, the XY-plane measures it along two dimensions i.e. one horizontal line called X-axis and a vertical line called Y-axis. Just like a number line, the X and Y axes extend towards infinity in both directions. Just like number line each horizontal partition on X axis equals 1 unit and each vertical partition on Y axis equals 1 unit.
So what is an XY coordinate plane?
Well as they say a picture is worth a thousand words. Here is the picture that explains what an XY coordinate plane is:

The X-Y plane refers to the X and Y axis that are perpendicular lines intersecting at point called origin (Point marked as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} O$). The two axes act as reference points for the objects on the plane.  X axis is horizontal line and the Y axis the vertical line. Both the axis extend from -∞ to + ∞ in both directions. There are some interesting facts about X-Y plane that are worth remembering:
1. Each point on the plane has a value that corresponds to the X and Y axes. These values are known as thecoordinates of the point.
2. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as a signed distances from the origin. Normally the coordinates are represented as $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} P(x,y)$.  The coordinates of origin are $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} O(0,0)$.
3. A point lying in the first quadrant has a positive value for both X and Y coordinates i.e. point $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} P(x,y)$lies in the first quadrant only if $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x>0$and  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y>0$
4. A point lying in the second quadrant has a negative value for X coordinate but a positive value for the Y coordinate i.e. point $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} P(x,y)$lies in the second quadrant only if $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x<0$and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y>0$
5. A point lying in the third quadrant has a negative value for both X and Y coordinates i.e. point $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} P(x,y)$lies in the third quadrant only if $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x<0$and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y<0$
6. A point lying in the fourth quadrant has a positive value for X coordinate but a negative value for the Y coordinate i.e. point $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} P(x,y)$lies in the second quadrant only if $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x>0$and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y<0$
The XY plane can be used to graph a function or an equation between two variables (represented by $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$ and $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y$coordinates). The graph is the set of all points satisfying that function or equation. For example, the set of all points ( $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x$$\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y$) where  $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y=f(x)$ is the graph of the function $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} f$. For example the function $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y=f(x)=x-2$ can be plotted on the X-Y plane as follows:
As can be seen each and every point on the line shown in the figure above satisfies the equation $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} y=f(x)=x-2$
For Example:
 Point C (4,2) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x=4, y=2 \Rightarrow 2=4-2$ Point D(-1,-3) $\usepackage{color} \definecolor{Myblue}{rgb}{0.27,0.38,0.5} \color{Myblue} x=-1, y=-3 \Rightarrow -3=-1-2$

Example:
The above figure shows an XY-plane with points labeled A through E. Which of the following points has coordinates of (4,1)?
A) P
B) Q
C) R
D) S
E) T
Solution:

The coordinates (4,1) implies that x=4 and y=1. Since both x and y coordinates are positive, the points must be in Quadrant-I.
There are only 2 points: R and Q in first quadrant. If you draw perpendicular projections to X and Y axis from point Q, you will see that point Q has x coordinates of 4 and y coordinate of 1.
Therefore Point has coordinates of x=4, y=1