Functioning in the Real World Precalculus - Chapter 09, Angielskie [EN](1)
[ Pobierz całość w formacie PDF ]
gord.3896.09.pgs 4/24/03 10:15 AM Page 583
9
Geometric Models
9.1
Introduction to Coordinate Systems
What is a coordinate system? In simple terms, a coordinate system provides a way
to
locate
and
identify
points in the plane. In the usual
rectangular
or
Cartesian coor-
dinate system,
every point can be pictured in either of two ways. First, a point
P
with coordinates can be thought of as lying at the corner of a unique rec-
tangle whose opposite corner is at the origin and two of whose sides lie along the
two coordinate axes, as illustrated in Figure 9.1(a). The base of this rectangle is
and its height is Second, the point
P
can be thought of as the intersection of two
perpendicular lines, one parallel to the
y
-axis at a distance of
1
x
0
,
y
0
2
x
0
,
y
0
.
x
0
from it and the
other parallel to the
x
-axis at a height of
y
0
from it, as illustrated in Figure 9.1(b).
y
y
P
(
x
0
,
y
0
)
y
=
y
0
P
(
x
0
,
y
0
)
y
0
x
=
x
0
x
x
x
0
(a)
(b)
FIGURE 9.1
Mathematicians have found that, in many situations, rectangular coordinates
are not the most natural or the most effective way to locate points and have devel-
oped alternative coordinate systems. One such approach involves the use of two
axes that are not perpendicular, but rather meet at the origin at some angle other
than a right angle. Points in such a slanted coordinate system can be located at the
opposing vertex of a parallelogram, as illustrated in Figure 9.2.
583
gord.3896.09.pgs 4/24/03 10:15 AM Page 584
584
CHAPTER 9
Geometric Models
v
P
(
u
0
,
v
0
)
v
0
u
0
u
FIGURE 9.2
Another approach is to locate a point by using a circle centered at the origin
O
instead of a rectangle. To do so requires specifying both the radius of the circle and
an angle to indicate where on the circle the point is located. This approach leads
to the
polar coordinate system,
which is illustrated in Figure 9.3. We investigate
this coordinate system in Sections 9.6 and 9.7.
u
P
(
r
, )
θ
r
θ
O
FIGURE 9.3
Other approaches are used for particular applications that involve locating
points lying on some ellipse centered at the origin (an elliptic coordinate system),
on some parabola (a parabolic coordinate system), or on a hyperbola (a hyperbolic
coordinate system), as illustrated in Figures 9.4(a–c), respectively. In fact, the long
range navigation (LORAN) system used by navigators in ships and planes to locate
their positions is based on the fact that every point in a plane can be interpreted as
lying at the intersection of two hyperbolas in a hyperbolic coordinate system.
y
y
y
x
x
x
FIGURE 9.4
In this chapter, we first develop several special characteristics of the rectangu-
lar coordinate system and then show how to represent some extremely important
curves. Later we explore other ways to represent functions.
gord.3896.09.pgs 4/24/03 10:15 AM Page 585
9.2
Analytic Geometry
585
9.2
Analytic Geometry
One of the most useful and far-reaching developments in mathematics is Rene
Descartes’s idea of representing algebraic concepts geometrically. This approach,
known as
analytic geometry,
lets you visualize the mathematics graphically to com-
plement the algebraic approach that is based on symbols. Everything we have done in-
volving graphs of functions is an outgrowth of Descartes’s ideas. In this section we
examine some additional ideas involving points, lines, and circles in the plane.
We begin by considering the two points
A
at and
B
at in the
plane. You already know how to find an equation of the line through them by using
either the point–slope form
1
x
0
,
y
0
2
1
x
1
,
y
1
2
y
y
0
m
1
x
x
0
2
or the slope–intercept form
y
mx
b
,
where the slope of the line is
y
1
y
0
m
x
0
.
x
1
Alternatively, we have the implicit form for the equation of a line,
ax
by
c
,
where
c
>
a
and
c
>
b
represent the
x
- and the
y
-intercepts of the line, respectively.
Distance Between Points
We now ask: What is the distance between the points
A
at and
B
at
We write this distance as Figure 9.5 shows that the points
A
and
B
determine
a right triangle
ABC
; the coordinates of point
C
are because
C
is at the same
horizontal distance as
B
(measured from the
y
-axis) and at the same vertical height
as
A
(measured from the
x
-axis). Moreover, the horizontal distance from
A
to
C
is
it is the change, or difference, in the
x
-coordinates. Similarly, the vertical
distance from
C
to
B
is it is the change in the
y
-coordinates. Consequent-
ly, the distance from
A
to
B
is the length of the hypotenuse of this right triangle.
The Pythagorean theorem therefore gives us the
distance formula.
1
x
0
,
y
0
2
1
x
1
,
y
1
2
?
AB
.
1
x
1
,
y
0
2
x
1
x
0
;
y
1
y
0
;
y
B
(
x
1
,
y
1
)
y
1
√
(
x
1
–
x
0
)
2
+ (
y
1
–
y
0
)
2
|
AB
| =
y
1
–
y
0
x
1
–
x
0
y
0
C
(
x
1
,
y
0
)
A
(
x
0
,
y
0
)
x
FIGURE 9.5
x
0
x
1
  gord.3896.09.pgs 4/24/03 10:15 AM Page 586
586
CHAPTER 9
Geometric Models
Distance Formula
The distance from point
A
at
1
x
0
,
y
0
2
to point
B
at
1
x
1
,
y
1
2
is
2
2
.
AB
2
1
x
1
x
0
2
1
y
1
y
0
2
E
XAMPLE 1
Find the distance from the point
A
at
1
2
to the point
B
at
1
2
2, 5
6, 8
.
Applying the distance formula gives
Solution
AB
2
1
2
2
1
2
2
6
2
8
5
2
2
16
19
25
5 units.
Consider again the two points
A
at and
B
at in the plane. Suppose
that we want to determine the
midpoint M
of the line segment connecting
A
to
B
.
Figure 9.6 shows that the points
A
and
B
determine a right triangle
ABC
and that the
points
A
and
M
determine a smaller right triangle
AMD
. These two right triangles
1
x
0
,
y
0
2
1
x
1
,
y
1
2
y
B
(
x
1
,
y
1
)
1
2
1
2
M
(
x
0
+ (
x
1
–
x
0
),
y
0
+
(
y
1
–
y
0
)
)
1
2
AB
1
2
CB
1
2
AC
C
(
x
1
,
y
0
)
D
(
x
0
+ (
x
1
–
x
0
),
y
0
)
1
2
A
(
x
0
,
y
0
)
x
FIGURE 9.6
are similar, and hence their corresponding sides are proportional (see Appendix A4).
Because
M
is halfway from
A
to
B
, we see that
D
is halfway from
A
to
C
, and the
height
DM
is half the height
CB
. Thus the
x
-coordinate at
D
(and hence also at
M
) is
1
2
x
x
0
1
x
1
x
0
2
.
Similarly, because the height
DM
is half the height
CB
, the
y
-coordinate at
M
is
1
2
y
y
0
1
y
1
y
0
2
.
We can rewrite these expressions as
1
2
1
2
x
1
1
2
x
0
1
2
x
0
1
x
1
x
0
2
x
0
1
x
1
x
0
2
and
1
2
1
2
y
1
1
2
y
0
1
2
y
0
1
y
1
y
0
2
y
0
1
y
1
y
0
2
.
 gord.3896.09.pgs 4/24/03 10:15 AM Page 587
9.2
Analytic Geometry
587
Thus the coordinates of the midpoint
M
of a line segment are simply the averages
of the
x
-coordinates and the
y
-coordinates of the endpoints, respectively.
Midpoint Formula
The midpoint
M
of the line segment from
A
at
1
x
0
,
y
0
2
to
B
at
1
x
1
,
y
1
2
is at
1
2
1
2
x
x
0
1
x
1
x
0
2
,
y
y
0
1
y
1
y
0
2
or
y
1
y
0
x
1
x
0
x
,
y
.
2
2
E
XAMPLE 2
Find the midpoint of the line segment joining
A
at
1
1, 11
2
and
B
at
1
3, 7
2
.
The coordinates of the midpoint are
Solution
1
2
x
x
0
1
x
1
x
0
2
1
2
1
1
3
1
2
1
1
2
and
1
2
y
y
0
1
y
1
y
0
2
1
2
1
2
11
1
7
11
2
11
1
4
2
9.
Alternatively,
y
1
y
2
x
1
x
2
3
1
7
11
x
2
and
y
9.
2
2
2
2
Figure 9.7 shows the solution.
y
12
A
(1, 11)
10
M
(2, 9)
8
B
(3, 7)
6
4
2
x
FIGURE 9.7
0
1
2
3
4
We might also want to determine a point at some other fraction of the distance
from
A
to
B
. To do so, we simply extend the preceding argument to determine a
Â
[ Pobierz całość w formacie PDF ]