Precalculus Fifth Edition Mathematics for Calculus James Stewart  Lothar Redlin  Saleem Watson

1. Precalculus Fifth Edition Mathematics for Calculus James StewartLothar RedlinSaleem Watson

2. Fundamentals 1

3. Coordinate Geometry 1.8

4. Coordinate Geometry • The coordinate plane is the link between algebra and geometry. • In the coordinate plane, we can draw graphs of algebraic equations. • In turn, the graphs allow us to “see” the relationship between the variables in the equation.

5. The Coordinate Plane

6. The Coordinate Plane • Points on a line can be identified with real numbers to form the coordinate line. • Similarly, points in a plane can be identified with ordered pairs of numbers to form the coordinate plane or Cartesian plane.

7. Axes • To do this, we draw two perpendicular real lines that intersect at 0 on each line. • Usually, • One line is horizontal with positive direction to the right and is called the x-axis. • The other line is vertical with positive direction upward and is called the y-axis.

8. Origin & Quadrants • The point of intersection of the x-axis and the y-axis is the origin O. • Thetwo axes divide the plane into four quadrants, labeled I, II, III, and IV here.

9. Origin & Quadrants • The points onthe coordinate axes are not assigned to any quadrant.

10. Ordered Pair • Any point P in the coordinate plane can be located by a unique ordered pairof numbers (a, b). • The first number a is called the x-coordinateof P. • The second number b is called the y-coordinateof P.

11. Coordinates • We can think of the coordinates of P as its “address.” • They specify its location in the plane.

12. Coordinates • Several points are labeled with their coordinates in this figure.

13. E.g. 1—Graphing Regions in the Coordinate Plane • Describe and sketch the regions given by each set. • {(x, y) | x≥ 0} • {(x, y) | y = 1} • {(x, y) | |y| < 1}

14. Example (a) E.g. 1—Regions in the Coord. Plane • The points whose x-coordinates are 0 or positive lie on the y-axis or to the right of it.

15. Example (b) E.g. 1—Regions in the Coord. Plane • The set of all points with y-coordinate 1 is a horizontal line one unit above the x-axis.

16. Example (c) E.g. 1—Regions in the Coord. Plane • Recall from Section 1-7 that |y| < 1 if and only if –1 < y < 1 • So, the given region consists of those points in the plane whose y-coordinates lie between –1 and 1. • Thus, the region consists of all points that lie between (but not on) the horizontal lines y = 1 and y = –1.

17. Example (c) E.g. 1—Regions in the Coord. Plane • These lines are shown as broken lines here to indicate that the points on these lines do not lie in the set.

18. The Distance and Midpoint Formulas

19. The Distance Formula • We now find a formula for the distance d(A, B) between two points A(x1, y1) and B(x2, y2) in the plane.

20. The Distance Formula • Recall from Section 1-1 that the distance between points a and b on a number line is: d(a, b) = |b – a|

21. The Distance Formula • So, from the figure, we see that: • The distance between the points A(x1, y1) and C(x2, y1) on a horizontal line must be |x2 – x1|. • The distance between B(x2, y2) and C(x2, y1) on a vertical line must be |y2 – y1|.

22. The Distance Formula • Triangle ABC is a right triangle. • So, the Pythagorean Theorem gives:

23. Distance Formula • The distance between the points A(x1, y1) and B(x2, y2) in the plane is:

24. E.g. 2—Applying the Distance Formula • Which of the points P(1, –2) or Q(8, 9) is closer to the point A(5, 3)? • By the Distance Formula, we have:

25. E.g. 2—Applying the Distance Formula • This shows that d(P, A) < d(Q, A) • So, P is closer to A.

26. The Midpoint Formula • Now, let’s find the coordinates (x, y) of the midpoint M of the line segment that joins the point A(x1, y1) to the point B(x2, y2).

27. The Midpoint Formula • In the figure, notice that triangles APM and MQB are congruent because: • d(A, M) = d(M, B) • The corresponding angles are equal.

28. The Midpoint Formula • It follows that d(A, P) = d(M, Q). • So, x – x1 = x2 – x

29. The Midpoint Formula • Solving that equation for x, we get: 2x = x1 + x2 • Thus, x = (x1 + x2)/2 • Similarly, y = (y1 + y2)/2

30. Midpoint Formula • The midpoint of the line segment from A(x1, y1) to B(x2, y2) is:

31. E.g. 3—Applying the Midpoint Formula • Show that the quadrilateral with vertices P(1, 2), Q(4, 4), R(5, 9), and S(2, 7) is a parallelogram by proving that its two diagonals bisect each other.

32. E.g. 3—Applying the Midpoint Formula • If the two diagonals have the same midpoint, they must bisect each other. • The midpoint of the diagonal PR is: • The midpoint of the diagonal QS is:

33. E.g. 3—Applying the Midpoint Formula • Thus, each diagonal bisects the other. • A theorem from elementary geometry states that the quadrilateral is therefore a parallelogram.

34. Graphs of Equations in Two Variables

35. Equation in Two Variables • An equation in two variables, such as y = x2 + 1, expresses a relationship between two quantities.

36. Graph of an Equation in Two Variables • A point (x, y) satisfiesthe equation if it makes the equation true when the values for x and y are substituted into the equation. • For example, the point (3, 10) satisfies the equation y = x2 + 1 because 10 = 32 + 1. • However, the point (1, 3) does not, because 3 ≠ 12 + 1.

37. The Graph of an Equation • The graphof an equation in x and y is: • The set of all points (x, y) in the coordinate plane that satisfy the equation.

38. The Graph of an Equation • The graph of an equation is a curve. • So, to graph an equation, we: • Plot as many points as we can. • Connect them by a smooth curve.

39. E.g. 4—Sketching a Graph by Plotting Points • Sketch the graph of the equation • 2x – y = 3 • We first solve the given equation for y to get: y = 2x – 3

40. E.g. 4—Sketching a Graph by Plotting Points • This helps us calculate the y-coordinates in this table.

41. E.g. 4—Sketching a Graph by Plotting Points • Of course, there are infinitely many points on the graph—and it is impossible to plot all of them. • Still, the more points we plot, the better we can imagine what the graph represented by the equation looks like.

42. E.g. 4—Sketching a Graph by Plotting Points • We plot the points we found. • As they appear to lie on a line, we complete the graph by joining the points by a line.

43. E.g. 4—Sketching a Graph by Plotting Points • In Section 1-10, we verify that the graph of this equation is indeed a line.

44. E.g. 5—Sketching a Graph by Plotting Points • Sketch the graph of the equation y = x2 – 2

45. E.g. 5—Sketching a Graph by Plotting Points • We find some of the points that satisfy the equation in this table.

46. E.g. 5—Sketching a Graph by Plotting Points • We plot these points and then connect them by a smooth curve. • A curve with this shape is called a parabola.

47. E.g. 6—Graphing an Absolute Value Equation • Sketch the graph of the equation y = |x|

48. E.g. 6—Graphing an Absolute Value Equation • Again, we make a table of values.

49. E.g. 6—Graphing an Absolute Value Equation • We plot these points and use them to sketch the graph of the equation.

50. Intercepts