1. 1 Copyright © Cengage Learning. All rights reserved. 3 Functions and Graphs 3.1Rectangular Coordinate Systems

2. 2 Rectangular Coordinate Systems We shall now show how to assign an ordered pair (a, b) of real numbers to each point in a plane. Although we have also used the notation (a, b) to denote an open interval, there is little chance for confusion, since it should always be clear from our discussion whether (a, b) represents a point or an interval.

3. 3 Rectangular Coordinate Systems We introduce a rectangular, or Cartesian,  coordinate system in a plane by means of two perpendicular coordinate lines, called coordinate axes, that intersect at the origin O, as shown in Figure 1. Figure 1

4. 4 Rectangular Coordinate Systems We often refer to the horizontal line as the x-axis and the vertical line as the y-axis and label them x and y, respectively. The plane is then a coordinate plane, or an xy-plane. The coordinate axes divide the plane into four parts called the first, second, third, and fourth quadrants, labeled I, II, III, and IV, respectively (see Figure 1). Points on the axes do not belong to any quadrant. Each point P in an xy-plane may be assigned an ordered pair (a, b), as shown in Figure 1.

5. 5 Rectangular Coordinate Systems We call a the x-coordinate (or abscissa) of P, and b the y-coordinate (or ordinate). We say that P has coordinates (a, b) and refer to the point (a, b) or the point P (a, b). Conversely, every ordered pair (a, b) determines a point P with coordinates a and b. We plot a point by using a dot, as illustrated in Figure 2. Figure 2

6. 6 Rectangular Coordinate Systems We may use the following formula to find the distance between two points in a coordinate plane.

7. 7 Example 1 – Finding the distance between points Plot the points A(–3, 6) and B(5, 1), and find the distance d (A, B). Solution: The points are plotted in Figure 4. Figure 4

8. 8 Example 1 – Solution By the distance formula, d (A, B) = = = =  9.43. cont’d

9. 9 Example 4 – Finding a formula that describes a perpendicular bisector Given A(1, 7) and B(–3, 2), find a formula that expresses the fact that an arbitrary point P (x, y) is on the perpendicular bisector l of segment AB. Solution: By condition 2 of Example 3, P (x, y) is on l if and only if d (A, P) = d (B, P); that is,

10. 10 Example 4 – Solution To obtain a simpler formula, let us square both sides and simplify terms of the resulting equation, as follows: (x – 1) 2 + (y – 7) 2 = [x – (– 3)] 2 + (y – 2) 2 x 2 – 2x + 1 + y 2 – 14y + 49 = x 2 + 6x + 9 + y 2 – 4y + 4 –2x + 1 – 14y + 49 = 6x + 9 – 4y + 4 –8x – 10y = –37 8x + 10y = 37 cont’d

11. 11 Example 4 – Solution Note that, in particular, the last formula is true for the coordinates of the point C(4, ) in Example 3, since if x = 4 and y =, substitution in 8x + 10y gives us 8  4 + 10  = 37. cont’d

12. 12 Rectangular Coordinate Systems We can find the midpoint of a line segment by using the following formula.

13. 13 Rectangular Coordinate Systems To apply the midpoint formula, it may suffice to remember that the x-coordinate of the midpoint = the average of the x-coordinates, and that the y-coordinate of the midpoint = the average of the y-coordinates.

14. 14 Example 5 – Finding a midpoint Find the midpoint M of the line segment from P 1 (–2, 3) to P 2 (4, –2), and verify that d(P 1, M) = d(P 2, M). Solution: By the midpoint formula, the coordinates of M are

15. 15 Example 5 – Solution The three points P 1, P 2, and M are plotted in Figure 8. cont’d Figure 8

16. 16 Example 5 – Solution By the distance formula, d(P 1, M) = d(P 2, M) = Hence, d(P 1, M) = d(P 2, M). cont’d

17. 17 Rectangular Coordinate Systems The term graphing utility refers to either a graphing calculator or a computer equipped with appropriate software packages. The viewing rectangle of a graphing utility is the portion of the xy-plane shown on the screen. The boundaries (sides) of the viewing rectangle can be manually set by assigning a minimum x value (Xmin), a maximum x value (Xmax), the difference between the tick marks on the x-axis (Xscl), a minimum y value (Ymin), a maximum y value (Ymax), and the difference between the tick marks on the y-axis (Yscl).

18. 18 Rectangular Coordinate Systems In examples, we often use the standard (or default) values for the viewing rectangle. These values depend on the dimensions (measured in pixels) of the graphing utility screen. If we want a different view of the graph, we use the phrase “using [Xmin, Xmax, Xscl] by [Ymin, Ymax, Yscl]” to indicate the change in the viewing rectangle. If Xscl and/or Yscl are omitted, the default value is 1.