College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Coordinates and Graphs 2

3. Fundamentals • In Chapter 1, we solved equations and inequalities algebraically. • In the preceding section, we learned how to sketch the graph of an equation in a coordinate plane.

4. Graphing Calculators; Solving Equations and Inequalities Graphically 2.3

5. Solving Equations and Inequalities Graphically • In this section, we use graphs to solve equations and inequalities. • To do this, we must first draw a graph using a graphing device. • So, we begin by giving a few guidelines to help us use graphing devices effectively.

6. Using a Graphing Calculator

7. Viewing Rectangle • A graphing calculator or computer displays a rectangular portion of the graph of an equation in a display window or viewing screen. • We call this a viewing rectangle.

8. Viewing Rectangle • The default screen often gives an incomplete or misleading picture. • So, it is important to choose the viewing rectangle with care.

9. Viewing Rectangle • Let’s choose: • The x-values to range from a minimum value of Xmin = a to a maximum value of Xmax = b • The y-values to range from a minimum value of Ymin = c to a maximum value of Ymax = d.

10. Viewing Rectangle • Then, the displayed portion of the graph lies in the rectangle [a, b] x [c, d] = {(x, y) | a≤ x ≤ b, c ≤ y ≤ d} • We refer to this as the [a, b] by [c, d] viewing rectangle.

11. Using a Graphing Calculator • The graphing device draws the graph of an equation much as you would. • It plots points of the form (x, y) for a certain number of values of x, equally spaced between a and b.

12. Using a Graphing Calculator • If the equation is not defined for an x-value, or if the corresponding y-value lies outside the viewing rectangle, the device ignores this value and moves on to the next x-value. • It connects each point to the preceding plotted point to form a representation of the graph of the equation.

13. E.g. 1—Choosing an Appropriate Viewing Rectangle • Graph the equation y =x2 + 3 in an appropriate viewing rectangle. • Let’s experiment with different viewing rectangles. • We’ll start with the viewing rectangle [–2, 2] by [–2, 2] • So, we set: Xmin = –2 Ymin = –2 Xmax = 2 Ymax = 2

14. E.g. 1—Choosing an Appropriate Viewing Rectangle • The resulting graph is blank! • This is because x2≥ 0; so, x2 + 3 ≥ 3 for all x. • Thus, the graph lies entirely above the viewing rectangle. • So, this viewing rectangle is not appropriate.

15. E.g. 1—Choosing an Appropriate Viewing Rectangle • If we enlarge the viewing rectangle to [–4, 4] by [–4, 4], we begin to see a portion of the graph.

16. E.g. 1—Choosing an Appropriate Viewing Rectangle • If we try the viewing rectangle [–10, 10] by [–5, 30], we begin to get a more complete view of the graph.

17. E.g. 1—Choosing an Appropriate Viewing Rectangle • If we enlarge the viewing rectangle even further, the graph doesn’t show clearly that the y-intercept is 3.

18. E.g. 1—Choosing an Appropriate Viewing Rectangle • So, the viewing rectangle [–10, 10] by [–5, 30] gives an appropriate representation of the graph.

19. E.g. 2—Two Graphs on the Same Screen • Graph the equations y = 3x2 – 6x + 1 and y = 0.23x – 2.25 together in the viewing rectangle [1, –3] by [–2.5, 1.5] • Do the graphs intersect in this viewing rectangle?

20. E.g. 2—Two Graphs on the Same Screen • The figure shows the essential features of both graphs. • One is a parabola and the other is a line. • It looks as if the graphs intersect near the point (1, –2).

21. E.g. 2—Two Graphs on the Same Screen • However, if we zoom in on the area around this point, we see that, although the graphs almost touch, they don’t actually intersect.

22. Choosing a Viewing Rectangle • You can see from Examples 1 and 2 that the choice of a viewing rectangle makes a big difference in the appearance of a graph. • If you want an overview of the essential features of a graph, you must choose a relatively large viewing rectangle to obtain a global view of the graph. • If you want to investigate the details of a graph, you must zoom in to a small viewing rectangle that shows just the feature of interest.

23. Using a Graphing Calculator • Most graphing calculators can only graph equations in which y is isolated on one side of the equal sign. • The next example shows how to graph equations that don’t have this property.

24. E.g. 3—Graphing a Circle • Graph the circle x2 + y2 = 1. • We first solve for y—to isolate it on one side of the equal sign. y2 = 1 – x2 (Subtract x2) y = ± (Take square roots)

25. E.g. 3—Graphing a Circle • Thus, the circle is described by the graphs of two equations: • The first equation represents the top half of the circle (because y ≥ 0). • The second represents the bottom half (y ≤ 0).

26. E.g. 3—Graphing a Circle • If we graph the first equation in the viewing rectangle [–2, 2] by [–2, 2], we get the semicircle shown.

27. E.g. 3—Graphing a Circle • The graph of the second equation is the semicircle shown.

28. E.g. 3—Graphing a Circle • Graphing these semicircles together on the same viewing screen, we get the full circle shown.

29. Solving Equations Graphically

30. Solving Equations Algebraically • In Chapter 1, we learned how to solve equations. • To solve an equation like 3x – 5 = 0, we used the algebraic method. • This means we used the rules of algebra to isolate x on one side of the equation.

31. Solving Equations Algebraically • We view x as an unknown and we use the rules of algebra to hunt it down. • Here are the steps: 3x – 5 = 0 • 3x = 5 (Add 5) • x = 5/3 (Divide by 3)

32. Solving Equations Graphically • We can also solve this equation by the graphical method. • We view x as a variableand sketch the graph of the equation y = 3x – 5. • Different values for x give different values for y. • Our goal is to find the value of x for which y = 0.

33. Solving Equations Graphically • From the graph, we see that y = 0 when x ≈ 1.7. • Thus, the solution is x ≈ 1.7. • Note that, from the graph, we obtain an approximate solution.

34. Solving Equations Graphically • We summarize these methods here.

35. Algebraic Method—Advantages • The advantages of the algebraic method are that: • It gives exact answers. • The process of unraveling the equation to arrive at the answer helps us understand the algebraic structure of the equation.

36. Algebraic Method—Disadvantage • On the other hand, for many equations, it is difficult or impossible to isolate x.

37. Graphical Method—Advantages • The graphical method gives a numerical approximation to the answer. • This is an advantage when a numerical answer is desired. • For example, an engineer might find an answer expressed as x ≈ 2.6 more immediately useful than x = .

38. Graphical Method—Advantages • Also, graphing an equation helps us visualize how the solution is related to other values of the variable.

39. E.g. 4—Solving a Quadratic Equation • Solve the quadratic equations algebraically and graphically. • x2 – 4x + 2 = 0 • x2 – 4x + 4 = 0 • x2 – 4x + 6 = 0

40. Example (a) E.g. 4—Solving Algebraically • There are two solutions:

41. Example (b) E.g. 4—Solving Algebraically • There is just one solution, x = 2.

42. Example (c) E.g. 4—Solving Algebraically • There is no real solution.

43. E.g. 4—Solving Graphically • We graph the equations • y =x2 – 4x + 2y =x2 – 4x + 4y =x2 – 4x + 6 • By determining the x-intercepts of the graphs, we find the following solutions.

44. Example (a) E.g. 4—Solving Graphically • x ≈ 0.6 and x ≈ 3.4

45. Example (a) E.g. 4—Solving Graphically • x = 2

46. Example (c) E.g. 4—Solving Graphically • There is no x-intercept. • So, the equation has no solution.

47. Solving Quadratic Equations Graphically • The graphs in Figure 6 show visually why a quadratic equation may have two solutions, one solution, or no real solution. • We proved this fact algebraically in Section 1.3 when we studied the discriminant.

48. E.g. 5—Another Graphical Method • Solve the equation algebraically and graphically: 5 – 3x = 8x – 20

49. E.g. 5—Algebraic Solution • 5 – 3x = 8x – 20 • – 3x = 8x – 20 (Subtract 5) • –11x = –25 (Subtract 8x)

50. E.g. 5—Graphical Solution • We could: • Move all terms to one side of the equal sign. • Set the result equal to y. • Graph the resulting equation.