WOMEN IN SCIENCE—HARRIET BROOKS PITCHER
In a short career of 13 years, Harriet Brooks accomplished more as a physicist then did many
scientists in a lifetime. Obtaining her B.A. in mathematics and natural philosophy (the term used
for physics) in 1898 from McGill University, she went on to become the first woman to earn a
masters degree in 1901. She worked with giants in science, including Ernest Rutherford and
Marie Curie. She is best known for her work on radioactivity, conducting experiments with thorium
and with Marie Curie determining the atomic mass of radon. She became a faculty member of
Barnard College in 1904. In those days a university policy required married women to leave the
university. In 1907 Harriet Brooks married and had to resign.
This was a huge loss to science and Canadian research. She passed away 26 years later.
CANADIAN
TRADITION
MATHEMATICS
– J.C.
FIELDS
What she could have
accomplished IN
in those
years had she not been
barred
from research, we
Yournever
studyknow.
of Technical
follows a great
tradition
of excellence
mathematics
in
will
Harriet Mathematics
Brooks was considered
one of
the leading
women ofinher
times, second
Canada.
One Curie.
of the most renowned Canadian Mathematicians was John Charles Fields, born
only
to Marie
in Hamilton, Ontario in 1863. A great believer in mathematical research, Professor Fields was
instrumental in establishing the National Research Council Canada. The Fields name is known
around the world through the Fields Medal, often described as the “Nobel Prize” of mathematto
HD-4
HYDROPHOIL—ALEXANDER
GRAHAM
BELLby the International
four mathematicians
who must be under forty years old,
and are selected
Mathematical
Union.
AND
CASEY
BALDWIN
From 1885, Alexander Graham Bell, inventor of the telephone, spent summers with his family at
Beinn Bhreagh (pronounced Ben Bree-ah) near Baddeck, Cape Breton, Nova Scotia. He never actually
retired but used the resources from his telephone patents to continue inventing until his death in
1922. During WWI, he and Casey Baldwin developed a high-speed watercraft to allow the Canadian
and American navies to better patrol their coasts to protect shipping from enemy submarine attacks.
The war ended before his watercraft, a hydrofoil, was ready for production. Bell and Baldwin continued
development and produced the HD-4 hydrofoil, which in 1919 set a water speed record of 114 km/hr,
about 71 mi./hr. The record stood for 10 years.
5–1
CANADIAN BIOGRAPHY
CANADIAN BIOGRAPHY
CANADIAN INNOVATION
Mapping Rectangular Coordinates
The Rectangular Coordinate System
In Chapter 1 we plotted numbers on the number line. Suppose, now, that we take a second number line and place it at right angles to the first one, so that each intersects the other at the zero
mark, as in Fig. 5–1. We call this a rectangular coordinate system.
y
(+)
Quadrant
II
Quadrant
I
3
y axis
4
Rectangular coordinates are also
called Cartesian coordinates,
after the French mathematician
René Descartes (1596–1650).
Another type of coordinate
system we will use is called the
polar coordinate system.
2
1
x axis
(−)
−4
−3
−2
−1
0
−1
1
2
3
(+)
4
x
Origin
−2
Quadrant
III
−3
Quadrant
IV
−4
(−)
FIGURE 5–1 The rectangular
coordinate system.
135
136
Chapter 5
◆
Graphs
The horizontal number line is called the x axis, and the vertical line is called the y axis.
They intersect at the origin. These two axes divide the plane into four quadrants, numbered
counterclockwise, as in Fig. 5–1.
Graphing Ordered Pairs
Figure 5–2 shows a point P in the first quadrant. Its horizontal distance from the origin,
called the x coordinate or abscissa of the point, is 3 units. Its vertical distance from
the origin, called the y coordinate or ordinate of the point, is 2 units. The numbers in
the ordered pair (3, 2) are called the rectangular coordinates (or simply coordinates)
of the point. They are always written in the same order, with the x coordinate first
(i.e., in alphabetical order). The letter identifying the point is sometimes written before
the coordinates, as in P(3, 2).
To plot any ordered pair (h, k), simply place a point at a distance h units from the
y axis and k units from the x axis. Remember that negative values of x are located to
the left of the origin and that negative y values are below the origin.
y
3
P(3, 2)
Ordinate
(2 units)
2
1
0
1
2
Abscissa
(3 units)
FIGURE 5–2 Is it clear from
this figure why we call these
rectangular coordinates?
x
3
◆◆◆
Example 1: The points
P(4, 1)
Q(2, 3)
R(1, 2)
S(2, 3)
T(1.3, 2.7)
are shown plotted in Fig. 5–3.
Notice that the abscissa is negative in the second and third quadrants and that the ordinate
is negative in the third and fourth quadrants. Thus the signs of the coordinates of a point tell us
◆◆◆
the quadrant in which the point lies.
y
Q(−2, 3)
T(1.3, 2.7)
3
2
I
II
P(4, 1)
1
−2
−1
0
1
2
3
4
x
−1
R(−1, −2)
IV
−2
III
−3
S(2, −3)
FIGURE 5–3
◆◆◆ Example 2: The point (3, 5) lies in the third quadrant, for that is the only quadrant in
◆◆◆
which the abscissa and the ordinate are both negative.
Section 5–1
◆
Exercise 1
137
Mapping Rectangular Coordinates
◆
Rectangular Coordinates
Rectangular Coordinates
If h and k are positive quantities, in which quadrants would the following points lie?
2. (h, k)
1. (h, k)
3. (h, k)
4. (h, k)
5.
6.
7.
8.
Which quadrant contains points having a positive abscissa and a negative ordinate?
In which quadrants is the ordinate negative?
In which quadrants is the abscissa positive?
The ordinate of any point on a certain straight line is 5. Give the coordinates of the point
of intersection of that line and the y axis.
9. Find the abscissa of any point on a vertical straight line that passes through the point (7, 5).
y
3
Graphing Ordered Pairs
A
10. Write the coordinates of points A, B, C, and D in Fig. 5–4.
B
11. Write the coordinates of points E, F, G, and H in Fig. 5–4.
2
C
D
1
12. Write the coordinates of points A, B, C, and D in Fig. 5–5.
13. Write the coordinates of points E, F, G, and H in Fig. 5–5.
−3
−2
−1
F
G
0
A
FIGURE 5–4
B
1
E
2
3 x
C
D
−2
−3
FIGURE 5–5
14. Graph each point.
(a) (3, 5)
(d) (3.75, 1.42)
(b) (4, 2)
(e) (4, 3)
(c) (2.4, 3.8)
(f) (1, 3)
Graph each set of points, connect them, and identify the geometric figure formed.
15. (0.7, 2.1), (2.3, 2.1), (2.3, 0.5), and (0.7, 0.5)
16. (2, 12 ), (3, 112 ), (112 , 3), and (12 , 2)
17. (112 , 3), (212 , 12 ), and (12 , 12 )
−1
−3
−1
H
E
0
−2
2
−3
−1
F
y
3
1
−2
1
3 x
2
G
H
138
Chapter 5
◆
Graphs
18. (3, 1), (1, 12 ), (2, 3), and (4, 312 )
19. Three corners of a rectangle have the coordinates (4, 9), (8, 3), and (8, 1). Graphically
find the coordinate of the fourth corner.
5–2
Graphing a Function Using Ordered Pairs
If the function is given as a set of ordered pairs, simply plot each ordered pair. We usually
connect the points with a smooth curve unless we have reason to believe that there are sharp
corners, breaks, or gaps in the graph.
If the function is in the form of an equation, we obtain a table of ordered pairs by first
selecting values of x over the required domain and then computing corresponding values of y.
Since we are usually free to select any x values we like, we pick “easy” integer values. We then
plot the set of ordered pairs.
Our first graph will be of the straight line.
y
◆◆◆
Q
Solution: Substituting into the equation, we obtain
−1
3
Example 3: Graph the function y f (x) 2x 1 for values of x from 2 to 2.
2x
f (2) 2(2) 1 5
f (1) 2(1) 1 3
f (0) 2(0) 1 1
f (1) 2(1) 1 1
f (2) 2(2) 1 3
y=
2
x Intercept
−2
−1
1
0
−1
1
y Intercept
2
x
Rise = 6
−2
P
−3
In Sec. 5–3 we will discuss the straight line in more detail and show another way
to graph it.
Our next graph will be of a curve called the parabola. We graph it, and any other
function, in the same way that we graphed the straight line: Make a table of point pairs,
plot the points, and connect them.
Run = 3
−4
−5
FIGURE 5–6 A first-degree
equation will always plot as a
straight line—hence the name
linear equation.
Thus our points are (2, 5), (1, 3), (0, 1), (1, 1), and (2, 3). Note that each of
these pairs of numbers satisfies the given equation.
These points plot as a straight line (Fig. 5–6). Had we known in advance that the
graph would be a straight line, we could have saved time by plotting just two points,
◆◆◆
with perhaps a third as a check.
◆◆◆
Example 4: Graph the function y f (x) x2 4x 3 for values of x from 1 to 5.
Solution: Substituting into the equation, we obtain
f (1) (1)2 4(1) 3 1 4 3 2
f (0) 02 0 3 3
f (1) 12 4(1) 3 1 4 3 6
f (2) 22 4(2) 3 4 8 3 7
f (3) 32 4(3) 3 9 12 3 6
f (4) 42 4(4) 3 16 16 3 3
f (5) 52 4(5) 3 25 20 3 2
The points obtained are plotted in Fig. 5–7.
Common
Error
Be especially careful when substituting negative values into
an equation. It is easy to make an error.