.
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.
CARTESIAN COORDINATE SYSTEM
Objective:
.
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).
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