CARTESIAN COORDINATE SYSTEM

Objective:

• To plot points in the Cartesian plane.
• Find the distance between two points in the plane.
• Find the midpoint of a line segment.
Recall 1.  We have seen a geometry of the real numbers before.  The real numbers can be organized on a number line that contains the number 0 and orders numbers in such a way that the numbers increase positively as they appear to  the right of 0 and increase negatively as they appear to the left of 0. This is illustrated below

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This number line organizes the way we think about real numbers, just as a calendar    organizes the days of the year, or as a clock organizes the hours and minutes of a day.

Recall 2.  The function concept was studied in another topic.  A function from a set of real numbers to another set of real numbers needs some way of keeping track of the corresponding numbers. For example, if the function is  f(x) = x2, then we have

f(-3) = 9             f(-2) = 4
f(-sqrt(2)) = 2      f(-1) = 1
f(0) = 0              f(1) = 1
f(sqrt(2)) = 2       f(2) = 4
etc.

But this does not seem to be a promising way to organize the correspondence, especially when there are  an infinite number of values to put on this list. However, looking at the partial list above, we notice that we are pairing numbers-- 2 with 4, 3 with 9, etc.  What if we made a list of pairs in this way?    The list would look like:

(-3,9)              (-2,4)
(-sqrt(2),2)       (-1,1)
(0,0)               (1,1)
(sqrt(2),2)        (2,4)
etc.

Is there a geometric way of viewing this data, a way that is similar to how we view real numbers on the number line? You probably know the answer to this question.  A graph of the function is a geometric picture of the behavior of the function.

Recall 3. (The Cartesian Coordinate System)

All points on the real number line have an address, namely the distance from 0, measured positively if to the right of 0, negatively if to the left of 0. We may construct a generalization of the real number line, called the real number plane, where all points have an address, a location that is described by a pair of numbers --go such a distance to the left (or right), then go such a distance up (or down.)

Take two copies of the real number line to construct the plane, one horizontal and the other vertical.  The two lines are called the coordinate axes.  The point (0,0) at which they intersect is called the origin.  The  horizontal axis is usually called the x-axis, the vertical line the y-axis. Notice that any point in plane can be located by an address given as a pair of real  numbers. The point (3,2) is an illustration of such an  address. The set of all points that can be described in  this way is called the Cartesian coordinate plane(or, simply, the plane).

Note
Additional terminology:  The address, or the pair of numbers describing the position of the point in the plane, is called the Cartesian coordinates of the point.  If the address of a point is (a,b), then a is called the x-coordinate (or the abscissa) and b is called the y-coordinate (or the ordinate). The coordinate axes divide the plane into four regions, often called the 1-st, 2-nd, 3 rd and 4th quadrants if you start with the 1st quadrant in the upper right hand corner and count clockwise.

Recall 4. Just as every point in the plane has an address given by a pair of real numbers (a,b), so too does every pair of real numbers describe a point in the plane. If two  points in the plane have addresses given by (a,b)    and (c,d), respectively, then they must have a distance    between them.  The question to ask is whether or not    that distance can be calculated directly from the values    of the pairs of real numbers (a,b) and (c,d).  The answer  is that the distance can easily be calculated by the    formula that comes as a direct consequence and  application of the Pythagorean Theorem for right    triangles, which says that in a right triangle with sides A and B, the square of the hypotenuse C equals the sum    of the squares of A and B (also, any triangle satisfying this property is a right triangle).  In other words:  The  triangle illustrated below has a hypotenuse of 5 units and

We may use the Pythagorean Theorem to calculate the distance between any two points in the plane.   Here's how.  Draw a right triangle in such a way that the base is parallel to the x-axis, the height is parallel to the y-axis and the two points are connected by the    hypotenuse.  This is illustrated for the two points (2,3)  and (-3,-2).  The length of the base of the triangle is    measured along the x-axis as 5 units, while the length of  the height of the triangle is measured along the y-axis as    also 5 units.  By the Pythagorean Theorem we have the   square of the hypotenuse to be 5*5 +5*5, or 50.  So the hypotenuse has length  sqrt(50). The length of the hypotenuse is also the distance between the two points (2,3) and (-3,-2) in the Cartesian plane:

In general, the distance between two points (a,b) and  (c,d) is .
Notice that the (a - c) term is the length of the base of the right triangle, whose hypotenuse joins (a,b) to (c,d).  Similarly, the (b - d) term is the length of the height of such a triangle.

Example 1.    What geometric shape is outlined by the lines joining A to B, B to C, C to D and D to A, if
A = (-14,-5),  B = (-5,-12),  C = (18,5) and  D = (9,12)

Solution:    Make a sketch of the points on a sheet of graph paper or carefully draw the two coordinate axes of the Cartesian plane and mark the places of the four points A, B, C, and D.

The sketch suggests that the figure is a parallelogram.  Can we be sure?  One way to find out is to measure each of the pairs of opposite sides of the figure and check that these opposite sides are equal?  To measure these lengths, we use the distance formula on the lines AB and CD, then on the lines BC and DA. Here are the results:

Comparing these distances, we see that  AB=CD  and BC=DA, which signifies that the figure is indeed a parallelogram.

Example 2. What geometric shape is outlined by the lines joining A=(-4,14), B=(-14,-10) and C=(10,0)?

Solution:    Make a sketch of the points on a sheet of graph paper or carefully draw the two coordinate axes of the Cartesian plane and mark the places of the three points A, B, and C.

The sketch suggests that the figure is an isosceles triangle. Can we be sure? Measure each of the sides of the triangle that look equal and check that they are equal. To measure these lengths, we use the distance formula on the lines and, since they are the lines that look as though their lengths are equal.  Here are the results:

Comparing these distances, we see that AB=BC, as suspected. This means that the triangle is isosceles.

Example 3. Let f(x) = x*x.  Plot the points (-2,f(-2)), (-1,f(-1)), (0,f(0)), (1,f(1)), (2,f(2)) on the Cartesian plane and connect the points by consecutive straight lines.

Solution: First compute f(-2), f(-1), f(0), f(1) and f(2).

f(-2) = (-2)(-2) = 4,
f(-1) = (-1)(-1) = 1,
f(0) = (0)(0) = 0,
f(1) = (1)(1) = 1,
f(2) = (2)(2) = 4.

Now we have the values of the points that must be plotted.

(-2,4),   (-1,1),    (0,0),     (1,1),    and   (2,4).

We plot the points below.

Example 4. Let f(x) = (x - 1)2.  Plot the points (-1,f(-1)), (0,f(0)), (1,f(1)), (2,f(2)), (3,f(3)) on the Cartesian plane and connect the points by consecutive straight lines. Compare the sketch with the one from Example 3.

Solution: First compute f(-1), f(0), f(1), f(2), and f(3).

f(-1) = (-1 - 1)2 = 4
f(0) = (0 - 1)2 = 1,
f(1) = (1 - 1)2 = 0,
f(2) = (2 - 1)2 = 1,
f(3) = (3 - 1)2 = 4.

Now we have the values of the points that must be plotted.

(-1,4),   (0,1),    (1,0),     (2,1),    and   (3,4).

We plot the points below.

The figure that was sketched in Example 26.2 is dotted.  Note that the new figure is precisely the same as the dotted figure, only moved over to the right by one unit.  Also note that the new function f(x) = (x - 1)2  compares to the old function g(x) = x2 as g(x) = f(x + 1).  Notice the +1 comparison in the function and the shift of 1 unit to the right.

EXERCISES

1. Plot and label the following points (3,5),  (-2,6), (-5,-2),  (-9/2,5/2), (5.5,1.5) and (0,0).

2. Find the distance between the points (24,10) and (0,0).

3  Find the distance between (15,22) and (4,9).

4 Find the distance between (-23,-10) and (3,19).

5 Plot the points A = (1,3), B = (2,6), C = (4,7) and D = (3,4).  Connect these points by straight lines and determine what kind of a figure it outlines.

6 What kind of a figure is outlined by the straight lines that connect the points A = (2,5),  B = (3,7) and C = (0,4)?

7 What kind of figure is outlined by the points A = (6,1),  B = (2,-1), C = (4,5) and D = (0,3)?

8 Show that the points A = (1,3), B = (-2,8), and C = (2,7) are corners of a right triangle.

9 Sketch all points having x coordinate equal to 5.

10 Sketch all points having y coordinate equal to -3.

11 Sketch the region of points where -3<=x<=5 and -5<=y<=10.

12 Sketch the points given by the data in the table below.

13 Sketch the points given by the data in the table below.

14 Let f(x) = -x3.  Plot the points (-2,f(-2)),(-1,f(-1)), (0,f(0)), (1,f(1)), and (2,f(2)).   Sketch the straight lines joining these points.

15 Suppose that f(x) = (x-3)3.  Sketch the straight lines joining the points (1,f(1)),(2,f(2)), (3,f(3)), (4,f(4)), and (5,f(5)).

2  26 units

5  parallelogram

6  isosceles triangle

7  square