Section 3.1  Review - The Cartesian Coordinate System


The Cartesian Coordinate System was established by 18th century French mathematician Rene Descartes.  It gives us a method of creating a graph or picture of our ordered pairs.   Once established, we can use a graph to analyze our ordered pairs.

The Cartesian Coordinate System looks like a piece of graph paper.  At the center are two perpendicular number lines.  The point were these lines intersect is called the origin.   The coordinates of the origin is the ordered pair (0, 0).

graph

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Plotting Ordered Pairs

The two perpendicular lines that form the center of our Cartesian Coordinate System are called axis.  The horizontal line passing through the origin is called the x-axis.   The vertical line passing through the origin is called the y-axis.  To plot the ordered pair (x, y), move x units along the x-axis and y units along the y-axis.  

EXAMPLE 1:  Plot the ordered pair (2, 3) on the Cartesian Coordinate System.

To plot the ordered pair (2, 3), we need to move 2 units to the right on the x-axis and 3 units up on the y-axis. 

plot of (3, 2)

EXAMPLE 2:  Plot the ordered pair (-1, -2) on the Cartesian Coordinate System.

To plot the ordered pair (-1, -2), we need to move 1 units to the left on the x-axis and 2 units down on the y-axis. 

lot of (-1, -2)


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EXAMPLE 3:  On a piece of graph paper, plot the following ordered pairs?

A]  (5, 4)        B] (-2, 3)         c]  (1/3, 5/8)        d] (0, 0)        e]  (3, -2)

Answer 3:  To view the answers to the above example,

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Identifying Plotted Points

While plotting points on a graph is important, it is equally important to be able to read points from a graph.  To find the coordinates of a point,

1.  Starting from the origin, move along the x-axis to find the x-coordinate.

2.  Then moving along the y-axis, find the y-coordinate.

 

EXAMPLE 4:  Determine the coordinates of each of the points below.

Determine the coordinates

To determine the coordinates of point A, first move along the x-axis until we reach A.   Since we needed to move two units to the left of the origin, the x-coordinate is -2.  Since we do not have to move along the y-axis, the y-coordinate is 0.   Putting this together, the coordinates of the point A is (-2, 0).

To determine the coordinates of point B, first move along the x-axis until we reach B.   Since we do not have to move along the x-axis, the x coordinate is 0.  Now move along the y-axis until we reach B.  Since we need to move down two units, the y-coordinate is -2.  Putting this together, the coordinates of the point B is (0, -2).

Notice that the point (-2, 0) is not at the same position on the graph as the point (0, -2).  Remember that with ordered pairs, order matters!  The value of x comes first and the value of y comes second.

To determine the coordinates of point C, first move along the x-axis until we reach C.   Since we needed to move three units to the right of the origin, the x-coordinate is 3.  Now move along the y-axis until we reach C.  Since we need to move up two units, the y-coordinate is 2.  Putting this together, the coordinates of the point C is (3, 2).

To determine the coordinates of point D, first move along the x-axis until we reach D.   Since we needed to move two units to the right of the origin, the x-coordinate is 2.  Now move along the y-axis until we reach D.  Since we need to move up three units, the y-coordinate is 3.  Putting this together, the coordinates of the point C is (2, 3).

Results of determining the coordinates

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EXAMPLE 5:  For the graph below, determine the coordinates of each point?   Remember that the x-coordinate comes first, then the y-coordinate.

fig2-6.gif

Answer 5:  To view the answers to the above example,


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Quadrants

When looking that the Cartesian Coordinate System centered from the origin, we notice the graph is broken up into four smaller pieces.  These pieces are called quadrants.   Quadrants are labeled counter-clockwise with the upper-right quadrant being called Quadrant I.  See the figure below.

 graph with quadrants

If we examine the graph more carefully, we will notice that the values of both x and y are positive in the first quadrant.  The values of x are negative in the second quadrant and the values of y are positive in the second quadrant.  In the third quadrant, both x and y are negative.  Finally, in the fourth quadrant, x is positive but y is negative.

quadrants

EXAMPLE 6:  Determine which quadrant each of the following points are in.

a]  (-1, 3)        b]  (1.8, 3.4)         c]  (-1/2, - 1)         d]  (2, -1)

 

ANSWER 6:  a]  Since x is negative and y is positive, (-1, 3) must be located in quadrant II.
                      b]   Since both x and y are positive, (1.8, 3.4) must be located in quadrant I.
                      c]   Since both x and y are negative, (-1/2, -1) must be located in quadrant III.
                      d]   Since x is positive and y is negative, (2, -1) must be located in quadrant IV.

NOTE:  You could also plot these points on a graph and determine the same results.  

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EXAMPLE 7:  Determine which quadrant each of the following points are in.

a]  (2, 4)        b]  (1, -3)         c]  (2.2, 2.3)         d]  (-4, -5)         e]  (-1, 1)

Answer 7:  Points a and c are in quadrant I.  b is in quadrant IV.  d is in quadrant III and e is in quadrant II.  To view the answers to the above example,


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Scaling the Axes

Up until now, each unit on our graph has been represented by 1.  That is not always desirable.  Consider plotting a graph of the the data below.

YEAR ANNUAL
1960 60
1970 72
1980 83
1990 91
2000 85

Starting our graph at the origin would cause use to count 1960 places to the right and 60 places up before we could plot our first point!  It would be much easier if we could just start our x-axis at 1960 and start our y-axis at 60.  As long as we clearly mark our graph, we are allowed to do this.  Similarly, since our data is in multiples of 10, it makes sense to count by 10's along both axes instead of by 1's.  See the graph below.

Altered graph with x axis starting at 1960 and y axis starting at 60.  Counting in multiples of 10

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EXAMPLE 8:  For the graph below, determine the value of each ordered pair labeled A through C.

Reading coordinates based on the graph.

ANSWERS 8:  The coordinates for A are (150, 40).
                         The coordinates for B are (200, 60).
                         The coordinates for C are (250, 80).

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Noticing Trends and Making Predictions

Creating a graphs gives a picture of what a data set looks like.  Frequently we can use this picture to make estimates about data not readily seen in the chart.  Consider the chart above.  Notice that as the value of x increases, the value of y appears to be increasing as well. 

We can use this trend in the data to make a prediction about what the value of y will be based on any value of x between 150 and 250.  For example, when  x is 225, we can predict the value of y will be approximately 70.  How can we do that?  Let's look at the picture more closely.

fig2-9

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homework.gif First go to your textbook and read pages 169 through 172.  Then do questions 1 - 10 (all), on pages 180 - 181. 


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Copyright 2004 Donna Tupper.