Transcript section 1.1

Chapter 1.
Straight Lines and Linear Functions
• The Cartesian Coordinate System
• Straight Lines
• Linear Functions and Mathematical Models
• Intersection of Straight Lines
• The Method of Least Squares (Optional)
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
1.1 The Cartesian Coordinate System
• Real numbers may be represented geometrically by
points on a line. This line is called the real number line,
or coordinate line.
• We can construct the real number line as follows:
– Select the number 0 (such point is called the origin).
– Determine the scale.
– Each positive (negative) real number x lies x units to
the right (left) of 0.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Real Numbers
The real numbers can be ordered and represented in order on a
number line
0
-1.87
2
4.55
x
-3 -2 -1 0 1 2 3 4
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
•
•
•
•
•
Take two perpendicular lines
These lines intersect at 0, called the origin.
The horizontal line is called the x-axis
The vertical line is called the y-axis.
A number scale is set alone the x-axis, with positive
numbers lying to the right of the origin and negative
numbers lying to the left of the origin, similarly for the yaxis.
• The two number scales need not be the same.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
We can represent a point in the plane uniquely by an ordered
pair of numbers: (x, y)
• Give a point P, we can find an ordered pair (x, y)
corresponding to it by drawing perpendiculars from P to
the x-axis and y-axis.
• Conversely, given an ordered pair (x, y), we can located
the point P.
• In the ordered pair (x, y), x is called the x-coordinate, y is
called the y- coordinate.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
y
y-axis
(x, y)
x
x-axis
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Cartesian Coordinate System
y
Ex. Plot (4, 2)
Ex. Plot (-2, -1)
Ex. Plot (2, -3)
(4, 2)
x
(-2, -1)
(2, -3)
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Quadrants
The axes divide the plane into four quadrants.
• Quadrant I consists of P (x, y) with x>0 and y>0
• Quadrant II consists of P (x, y) with x<0 and y>0
• Quadrant III consists of P (x, y) with x<0 and y<0
• Quadrant IV consists of P (x, y) with x>0 and y<0
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Cartesian Coordinate System
Quadrants…
y
II
I
x
III
IV
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula
• The distance d between two points P1(x1, y1) and
P2(x2, y2) in the plane is given by
d
 x1  x2    y1  y2 
2
2
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula
d
 x1  x2    y1  y2 
2
2
y
 x2 , y2 
x
 x1, y1 
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Examples
Find the distance between (7, 5) and (-3, -2)
Solution
Let (7, 5) and (-3, -2) be points in the plane. We have
x1 = 7 y1 = 5 x2 = -3
y2 = -2
Using the distance formula, we have
d
 x1  x2    y1  y2 
d
 7  (3) 
2
2
2
  5  (2) 
d  100  49  149
2
The Equation of a Circle
A circle with center (h, k) and radius of length r can be
expressed in the form:
 x  h   y  k 
2
2
r
2
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Examples
Find an equation of the circle with radius 3 and center (-4, 1).
Solution
We use the circle formula with r = 3, h = -4, and k = 1:
 x  h   y  k 
2
2
 r2
 x  (4)   y  1  32
2
2
 x  4    y  1  9
2
2
x