1. 1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 2 Graphs and Functions

2. OBJECTIVES The Coordinate Plane Plot points in the Cartesian coordinate plane. Find the distance between two points. Find the midpoint of a line segment. SECTION 2.1 1 2 3 2 © 2010 Pearson Education, Inc. All rights reserved

3. Definitions A pair of real numbers in which the order is specified is called an ordered pair of real numbers. The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal if and only if x = a and y = b. The sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane. 3 © 2010 Pearson Education, Inc. All rights reserved

4. Definitions We begin with two coordinate lines, one horizontal (x-axis) and one vertical (y-axis), that intersect at their zero points. The point of intersection of the x-axis and y-axis is called the origin. The x-axis and y-axis are called coordinate axes, and the plane they determine is sometimes called the xy-plane. The axes divide the plane into four regions called quadrants, which are numbered as shown in the next slide. The points on the axes themselves do not belong to any of the quadrants. 4 © 2010 Pearson Education, Inc. All rights reserved

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6. Definitions The figure shows how each ordered pair (a, b) of real numbers is associated with a unique point in the plane P, and each point in the plane is associated with a unique ordered pair of real numbers. The first component, a, is called the x- coordinate of P and the second component, b, is called the y-coordinate of P, because we have called our horizontal axis the x-axis and our vertical axis the y-axis. 6 © 2010 Pearson Education, Inc. All rights reserved

7. Definitions The signs of the x- and y-coordinates are shown in the figure for each quadrant. We refer to the point corresponding to the ordered pair (a, b) as the graph of the ordered pair (a, b) in the coordinate system. The notations P(a, b) and P = (a, b) designate the point P in the coordinate plane whose x-coordinate is a and whose y-coordinate is b. 7 © 2010 Pearson Education, Inc. All rights reserved

8. EXAMPLE 1 Graphing Points Graph the following points in the xy-plane: Solution 3 units right, 1 unit up 3 units left, 4 units down2 units left, 4 units up3 units left, 0 units up or down2 units right, 3 units down 8 © 2010 Pearson Education, Inc. All rights reserved

9. EXAMPLE 1 Graphing Points Solution continued 9 © 2010 Pearson Education, Inc. All rights reserved

10. EXAMPLE 2 Plotting Data on Adult Smokers in the United States The data in table below show the prevalence of smoking among adults aged 18 years and older in the United States over the years 2000–2007. Plot the graph of the ordered pairs (year, %), where the first coordinate represents a year and the second coordinate represents the percent of adult smokers in that year. 10 © 2010 Pearson Education, Inc. All rights reserved

11. EXAMPLE 2 Plotting Data on Adult Smokers in the United States Solution (scatter diagram) 11 © 2010 Pearson Education, Inc. All rights reserved

12. TWO OTHER METHODS FOR DISPLAYING DATA 12 © 2010 Pearson Education, Inc. All rights reserved

13. TWO OTHER METHODS FOR DISPLAYING DATA 13 © 2010 Pearson Education, Inc. All rights reserved

14. THE DISTANCE FORMULA IN THE COORDINATE PLANE Let P(x 1, y 1 ) and Q(x 2, y 2 ) be any two points in the coordinate plane. Then the distance between P and Q, denoted d(P,Q), is given by the distance formula: 14 © 2010 Pearson Education, Inc. All rights reserved

15. THE DISTANCE FORMULA IN THE COORDINATE PLANE 15 © 2010 Pearson Education, Inc. All rights reserved

16. EXAMPLE 3 Find the distance between the points P(–2, 5) and Q(3, – 4). Finding the Distance Between Two Points Solution Let (x 1, y 1 ) = (–2, 5) and (x 2, y 2 ) = (3, – 4). 16 © 2010 Pearson Education, Inc. All rights reserved

17. EXAMPLE 4 Let A(4, 3), B(1, 4) and C(–2, – 4) be three points in the plane. a.Sketch the triangle ABC. b.Find the length of each side of the triangle. c.Show that ABC is a right triangle. Identifying a Right Triangle 17 © 2010 Pearson Education, Inc. All rights reserved

18. EXAMPLE 4 Solution Identifying a Right Triangle a. Sketch the triangle ABC. 18 © 2010 Pearson Education, Inc. All rights reserved

19. Solution continued b. Find the length of each side of the triangle. EXAMPLE 4 Identifying a Right Triangle 19 © 2010 Pearson Education, Inc. All rights reserved

20. Solution continued Check that a 2 + b 2 = c 2 holds in this triangle, where a, b, and c denote the lengths of its sides. The longest side, AC, has length 10 units. It follows from the converse of the Pythagorean Theorem that the triangle ABC is a right triangle. c. Show that ABC is a right triangle. EXAMPLE 4 Identifying a Right Triangle 20 © 2010 Pearson Education, Inc. All rights reserved

21. EXAMPLE 5 The baseball “diamond” is in fact a square with a distance of 90 feet between each of the consecutive bases. Use an appropriate coordinate system to calculate the distance the ball will travel when the third baseman throws it from third base to first base. Applying the Distance Formula to Baseball Solution We can conveniently choose home plate as the origin and place the x-axis along the line from home plate to first base and the y-axis along 21 © 2010 Pearson Education, Inc. All rights reserved

22. EXAMPLE 5 Solution continued Applying the Distance Formula to Baseball the line from home plate to third base. The coordinates of home plate (O), first base (A) second base (C) and third base (B) are shown. 22 © 2010 Pearson Education, Inc. All rights reserved

23. EXAMPLE 5 Find the distance between points A and B. Solution continued Applying the Distance Formula to Baseball 23 © 2010 Pearson Education, Inc. All rights reserved

24. THE MIDPOINT FORMULA The coordinates of the midpoint M(x, y) of the line segment joining P(x 1, y 1 ) and Q(x 2, y 2 ) are given by 24 © 2010 Pearson Education, Inc. All rights reserved

25. THE MIDPOINT FORMULA 25 © 2010 Pearson Education, Inc. All rights reserved

26. EXAMPLE 6 Find the midpoint of the line segment joining the points P(–3, 6) and Q(1, 4). Finding the Distance Between Two Points Solution Let (x 1, y 1 ) = (–3, 6) and (x 2, y 2 ) = (1, 4). 26 © 2010 Pearson Education, Inc. All rights reserved