1
Math 71 Outline Lecture Notes
(Prepared by Stefan Waner)
1. The Cartesian Plane and Distance
The
coordinate plane
is the infinite plane with
two perpendicular axes—the
x
-axis and the
y
-axis. The axes divide the plane into four
quadrants—shown in class. To locate a point, we use coordinates, so that each point is
represented by a pair
(a, b)
of real numbers.
Example
A. Locate the points
P(2,3), Q(-4,3), R(-5,-2), S(4,-3), T(0,-3)
and
U(4,0)
B. Sketch the following regions:
x = 4; |x|
≥
5, x+y
≥
1.
Distance Between Two Points
To find the distance between the general points P
1
(x
1
,
y
1
)
and P
2
(
x
2
, y
2
), introduce the third point Q
(x
2
, y
1
)
as shown, and use Pythagoras' Theorem
(shown in class). We obtain the formula:
d
2
= (x
2
- x
1
)
2
+ (y
2
- y
1
)
2
giving
d =
(x
2
- x
1
)
2
+ (y
2
- y
1
)
2
Distance Formula
Examples
A. Find the distance between
P(1,2)
and
Q(-1,6)
.
B. Find the distance from the general point
(a, b)
to the origin—gives another very
important formula.
2. Equations and Graphs
An
equation
is a mathematical expression containing an
“equals” sign and satisfying various syntactical rules. Eg.
x+y = 8
;
x
= 5, 1 = 0. Given
an equation in
x
and
y
, we can represent it by a “curve” in the coordinate plane. Formally:
The
graph
of an equation consists of all points
(x, y)
whose coordinates satisfy the
equation. To say that the coordinates of a point “satisfy” the equation means that when
you substitute them as values for x and y, the equation expresses a truth about numbers.
Examples
A. Graph the equation
x+y = 6.
B. Graph the equation
y = x
2
.
C. Graph the equation
x
2
+y
2
= 9.
D. Obtain the equation of a general circle center
(a, b)
radius r as:
(x - a)
2
+ (y - b)
2
= r
2
Circle Center
(a, b)
Radius
r
E. Sketch the graph of the equation
x
2
+ y
2
+ 2x - 6y + 7 = 0
.
F. Find the equation of the upper semicircle with radius 2 centered at
(3, -1).
3. Straight Lines
The
slope
of a straight line is a measure of its steepness, and is defined
as the number of units it goes up for every unit across (in the +x direction). In terms of a
formula,
m =
y
2
- y
1
x
2
- x
1
=
∆
y
∆
x
Slope Formula
given two points
(x
1
, y
1
)
and
(x
2
, y
2
)
on that line.
Example
Find the slope of the line through (1,2) and (2,1).