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) 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.

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 -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 y-axis. The two number scales need not be the same. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

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-axis (x, y) x x-axis Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Cartesian Coordinate System 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.

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.

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 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The Distance Formula y x 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 12

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