1. 1.9 Graphing Calculators: Solving Equations and Inequalities Graphically

2. 2 Objectives ► Using a Graphing Calculator ► Solving Equations Graphically ► Solving Inequalities Graphically

3. 3 Using a Graphing Calculator

4. 4 A graphing calculator or computer displays a rectangular portion of the graph of an equation in a display window or viewing screen, which we call a viewing rectangle. The default screen often gives an incomplete or misleading picture, so it is important to choose the viewing rectangle with care.

5. 5 Using a Graphing Calculator If we choose the x-values to range from a minimum value of Xmin = a to a maximum value of Xmax = b and the y-values to range from a minimum value of Ymin = c to a maximum value of Ymax = d, then the displayed portion of the graph lies in the rectangle [a, b]  [c, d] = {(x, y) | a  x  b, c  y  d} as shown in Figure 1. The viewing rectangle [a, b] by [c, d]

6. 6 Example 1 – Choosing an Appropriate Viewing Rectangle Graph the equation y = x 2 + 3 in an appropriate viewing rectangle.

7. 7 Solving Equations Graphically

8. 8 To solve an equation such as 3x – 5 = 0 we used the algebraic method. This means that we used the rules of algebra to isolate x on one side of the equation. We view x as an unknown, and we use the rules of algebra to hunt it down.

9. 9 Solving Equations Graphically We can also solve this equation by the graphical method. In this method we view x as a variable and 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.

10. 10 Solving Equations Graphically From the graph, we can see that y = 0 when x  1.7. Thus, the solution is x  1.7.

11. 11 Solving Equations Graphically

12. 12 Solving Equations Graphically The advantage of the algebraic method is that it gives exact answers. Also, the process of unraveling the equation to arrive at the answer helps us to understand the algebraic structure of the equation. On the other hand, for many equations it is difficult or impossible to isolate x. The graphical method gives a numerical approximation to the answer. This is an advantage when a numerical answer is desired.

13. 13 Example 4 – Solving a Quadratic Equation Algebraically and Graphically Solve the quadratic equations algebraically and graphically: (a) x 2 – 4x + 2 = 0 (b) x 2 – 4x + 4 = 0 (c) x 2 – 4x + 6 = 0

14. 14 Example 4 – Solving a Quadratic Equation Algebraically and Graphically

15. 15 Example 4 – Solving a Quadratic Equation Algebraically and Graphically Graphical Solution: y = x 2 – 4x + 2 y = x 2 – 4x + 4 y = x 2 – 4x + 6

16. 16 Solving Inequalities Graphically

17. 17 Solving Inequalities Graphically Inequalities can be solved graphically. To describe the method, we solve x 2 – 5x + 6  0 To solve the inequality graphically, we draw the graph of y = x 2 – 5x + 6 Our goal is to find those values of x for which y  0.

18. 18 Solving Inequalities Graphically These are simply the x-values for which the graph lies below the x-axis. We see that the solution of the inequality is the interval [2, 3]. x 2 – 5x + 6  0

19. 19 Example 8 – Solving an Inequality Graphically Solve the inequality: 3.7x 2 + 1.3x – 1.9  2.0 – 1.4x

20. 20 Example 8 – Solving an Inequality Graphically

21. 21 Graphing Calculators Practice: p. 104-105 #1-4, 9, 13, 17, 21, 27, 31, 38-39, 41, 47, 49, 53, 55, 59, 65, 67, 71-72, 77