1. Straight Lines and Linear Functions
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)
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2. 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.
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3. 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
-1.87
4.55 x
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4. 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 y-axis.
The two number scales need not be the same.
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5. 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.
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6. Cartesian Coordinate System
y-axis
(x, y) x
x-axis
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7. Cartesian Coordinate System
Ex. Plot (4, 2)
Ex. Plot (-2, -1)
Ex. Plot (2, -3)
(4, 2) x
(-2, -1)
(2, -3)
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8. 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
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9. Cartesian Coordinate System
Quadrants… y II I x
III IV
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10. 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
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11. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The Distance Formula y x
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12. 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 12

13. The Equation of a Circle
A circle with center (h, k) and radius of length r can be expressed in the form:
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14. 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 14