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Chapter 2.1
Graphs of Equations
The idea of pairing one quantity with another is
often encountered in everyday life. For example, a
numerical grade in a mathematics course is paired
with a corresponding letter grade. the number of
gallons of gasoline pumped into a tank is paired
with the amount of money needed to purchase it.
Another example is shown in the table, which
gives the dollars the average American spent in
2001 on entertainment. For each type of
entertainment, there is a corresponding number of
dollars spent.
Pairs of related quantities, such as a 96
determining a grade of A, 3 gallons of gasoline
costing $5.25, and 2001 spending on CDs of $47,
can be expressed as ordered pairs:
(96, A), (3, $5.25), (CDs, $47).
An ordered pair consists of two components,
written inside parentheses, in which the order of
the components is important.
Example 1 Writing Ordered Pairs
Use the table to write ordered pairs to express the
relationship between each type of entertainment
and the amount of money spent on it.
(a) DVD rentals/sales
Example 1 Writing Ordered Pairs
Use the table to write ordered pairs to express the
relationship between each type of entertainment
and the amount of money spent on it.
(b) movie tickets
In mathematics, we are most often interested in
ordered pairs whose components are numbers.
Note that (4,2) and (2,4) are different ordered
pairs because the order of the numbers is different.
The Rectangular Coordinate System.
As mentioned in Chapter R, each real number
corresponds to a point n a number line. This idea
is extended to ordered pairs of real numbers by
using tow perpendicular number lines, one
horizontal and one vertical, that intersect at their
zero-points.
This point of intersection is called the origin. The
horizontal line is called the x-axis, and the vertical
line is called the y-axis. Starting at the origin, on
the x-axis the positive numbers go to the right and
the negative numbers go to the left. The y-axis has
positive numbers going up and the negative
numbers going down.
The x-axis and y-axis together make up a
rectangular coordinate system, or Cartesian
coordinate system (names for one of its
coinventors, Rene Descartes; the other coinvertor
was Pierre de Fermat). The plane into which the
coordinate system is introduced is the coordinate
plane, or xy-plane.
The x-axis and y-axis divide the plane into four
regions, or quadrants, labeled as shown in Figure
1.
The points on the x-axis and y-axis belong to no
quadrant.
y
P (a,b)
y - axis
Quadrant Quadrant
II
I
Origin
x
Quadrant Quadrant
IV
III
x - axis
Each point P in the xy-plane corresponds to a
unique ordered pair (a,b) of real numbers. The
numbers a and b are the coordinates of point P. To
locate on the xy-plane the point corresponding to
the ordered pair (3,4), for example, start at the
origin, move 3 units in the positive x-direction,
and then move 4 units in the positive y-direction.
Point A corresponds to the ordered pair (3,4).
Also in figure 2, B corresponds to the ordered pair
(-5,6), C to (-2, -4), d TO (4,3), and E to (-3,0).
The point P corresponding to the ordered pair (a,b)
often is written P(a,b) as in Figure 1 and referred
tp as “the point (a,b).”
The Distance Formula
Recall that the distance on a number line between
the points P and Q with coordinates x1 and x2 is
d(P,q) = |x2 – x1| = |x1 – x2|
By using the coordinates of their ordered pairs, we
can extend this idea to find this distance between
any two points in a plane.
Figure 3 shows the points P(-4, 3) and R(8, -2)
To find the distance between these points we
complete a right triangle as in the figure. This
right triangle has its 900 angel at Q(8, 3). The
horizontal side of the triangle has length
d(P, Q) = |8 – (-4)| = 12
The vertical side of the triangle has length
d(Q, R) = |3 – (-2)| = 5
By the Pythagorean theorem, the length of the
remaining side of the triangle is
12  5
2
2
 144  25  169  13
This the distance between (-4, 3) and (8, -2) is 13
To obtain a general formula for the distance
between two points in a coordinate plane, let
P(x1, y1) and R (x2, y2) be any two distinct points
in a plane, as shown in Figure 3. Complete a
triangle by locating point Q with coordinates (x2,
y1).
d P, R 
x2  x1    y2  y1 
2
2
The Pythagorean theorem gives the distance
between P and R as
d P, R 
x2  x1    y2  y1 
2
2
Distance Formula
Suppose that P(x1, y1) and R (x2, y2) are two points
in a coordinate plane. The distance between P and
R, written d(P, R) is given by the distance formula,
d P, R 
x2  x1    y2  y1 
2
2
Example 2 Using the Distance Formula
Find the distance between P(-8, 4) and Q(3, -2)
Example 3 Determine Whether Three Points Are the Vertices of a Right
Triangle
Are points M(-2, 5), N(12, 3) and Q(10, -11) the
vertices of a right triangle?
d M , N  
          
d M , Q 
          
d N , Q  
          
2
2
2
2
2
2
Example 4 Determine Whether Three Points Are Colllinear
Are points M(-1, 5), N(2, -4) and Q(4, -10) collinear?
d M , N  
          
d M , Q 
          
d N , Q  
          
2
2
2
2
2
2
Midpoint Formula
The midpoint of the line segment with endpoints
(x1, y1) and (x2, y2) is
 x1  x2 y1  y2 
,


2 
 2
Example 5 Using the Midpoint Formula
Find the midpoint of the segment with endpoints
(8, -4) and (-6, 1).


    ,     
2
2

Example 6 Applying the Midpoint Formula to Data
Figure 8 depicts how the number of McDonald
restaurants worldwide increased from 1995 to 2001.
Use the midpoint formula and the two given points to
estimate the number of restaurants in 1998, and
compare it to the actual (rounded) figure of 24,000.


 
2
, 
 
2


Example 6 Applying the Midpoint Formula to Data
Figure 8 depicts how the number of McDonald
restaurants worldwide increased from 1995 to 2001.
Use the midpoint formula and the two given points to
estimate the number of restaurants in 1998, and
compare it to the actual (rounded) figure of 24,000.
 1995  2001 18000   30000  
,


2
2

Example 7 Finding Ordered Pairs That Are Solutions of Equations
For each equation, find three ordered pairs that are
solutions.
y  4x 1
Example 7 Finding Ordered Pairs That Are Solutions of Equations
For each equation, find three ordered pairs that are
solutions.
x  y 1
Example 7 Finding Ordered Pairs That Are Solutions of Equations
For each equation, find three ordered pairs that are
solutions.
x  y 1
Example 7 Finding Ordered Pairs That Are Solutions of Equations
For each equation, find three ordered pairs that are
solutions.
x  y 9
2
2
Example 8 Graph each equation of Example 7.
y
y  4x 1
x
Example 8 Graph each equation of Example 7.
y
x  y 1
x
Example 8 Graph each equation of Example 7.
y
x  y 9
2
x
2
Example 9 Finding the Equation of a Circle
Find and equation for the circle having radius 6 and
center at (-3, -4)
6
x     y   
2
2
Center-Radius Form of the Equation of a Circle
The circle with center (h, k) and radius r has the
equation
x  h    y  k 
2
2
r
2
The center-radius form of the equation of a circle.
A circle with center (0, 0) and radius r has
equation
2
2
2
x y r
Example 10 Graphing a Circle
Graph the circle
with equation
(x+3)2 + (y-4)2 = 36
y
x
Example 11 Finding the Center and Radius by Completing the Square
Decide whether or not each equation has a circle as its graph.
x2 – 6x + y2 + 10y + 25 = 0
Example 11 Finding the Center and Radius by Completing the Square
Decide whether or not each equation has a circle as its graph.
x2 + 10x + y2 – 4y + 33 = 0
Example 11 Finding the Center and Radius by Completing the Square
Decide whether or not each equation has a circle as its graph.
2x2 + 2y2– 6x + 10y = 1
Homework
Section 2.1 # 1 - 64