2.3. Graphing Calculators; Solving Equations and Inequalities Graphically

Transcription

1 2.3 Graphing Calculators; Solving Equations and Inequalities Graphically

2 Solving Equations and Inequalities Graphically To do this, we must first draw a graph using a graphing device, this is your TI-83/84 calculator. You should know how to use it to graph several equations at once Y=, WINDOW, ZOOM, TRACE, GRAPH buttons You should also know the second function of each of these buttons, particularly FORMAT, CALC and TABLE

3 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, find it by selecting WINDOW

4 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.

5 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.

6 Two Graphs on the Same Screen Graph the equations y = 3x 2 6x + 1 and y = 0.23x 2.25 together in the viewing rectangle [1, 3] by [ 2.5, 1.5] Enter on of these above as y 1 and the other as y 2 Press the GRAPH button to see both of these equations Do the graphs intersect in this viewing rectangle?

7 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).

8 Two Graphs on the Same Screen Use the ZOOM button to ZOOM in or create a ZOOMbox around the point to see what is really happening

9 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.

10 Graphing a Circle Graph the circle x 2 + y 2 = 1. We first solve for y to isolate it on one side of the equal sign. y 2 = 1 x 2 (Subtract x 2 ) y = ± (Take square roots)

11 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).

12 Graphing a Circle If we graph the first equation in the viewing rectangle [ 2, 2] by [ 2, 2], we get the semicircle shown. The graph of the second equation is the semicircle shown. Graphing these semicircles together on the same viewing screen, we get the full circle shown.

13 Solving Equations Graphically

14 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.

15 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)

16 Solving Equations Graphically We can also solve this equation by the graphical 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.

17 Solving Equations Graphically From the graph, we see that y = 0 when x 1.7. The solution is x 1.7.

18 Solving Equations Graphically We summarize these methods here.

19 Solving a Quadratic Equation 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

20 Solving Algebraically There are two solutions:

21 Solving Algebraically There is just one solution, x = 2.

22 Solving Algebraically There is no real solution.

23 Solving Graphically Now, see the power of graphing the equations as simultaneous equations in your calculator y = x 2 4x + 2 y = x 2 4x + 4 y = x 2 4x + 6 By determining the x-intercepts of the graphs, we find the following solutions. In our calculator, this is found when you do the CALC function which the second TRACE

24 Solving Graphically Example (a) x 0.6 and x 3.4

25 Solving Quadratic Equations Graphically There is no x-intercept for that last one, you can see it does not cross the x-axis therefore the equation has no solution. The graphs in Figure 6 show visually why a quadratic equation may have two solutions, one solution, or no real solution.

26 Solving Equations Graphically In the next example, we use the graphical method to solve an equation that is extremely difficult to solve algebraically.

27 E.g. 6 Solving an Equation in an Interval Solve the equation in the interval [1, 6]. We need to find all solutions x that satisfy 1 x 6. So, we will graph the equation in a viewing rectangle for which the x-values are restricted to this interval.

28 Solving an Equation in an Interval The figure shows the graph of the equation in the viewing rectangle [1, 6] by [ 5, 5]. There are two x-intercepts in this rectangle.

29 Solving an Equation in an Interval To find the x-intercepts on your calculator Use CALC, select 2 for zero Put the cursor before and after each point and put your guess around the intercept Do you see why you need to do this twice? 1 2

30 Solving an Equation in an Interval Zooming in, we see that the solutions are: x and x 3.72

31 Solving Inequalities Graphically

32 Solving Inequalities Graphically 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.

33 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].

34 Solving an Inequality Graphically Solve the inequality 3.7x x x Ouch, hard algebraically but pretty easy on the calculator graphically

35 Solving an Inequality Graphically We graph the equations in the same viewing rectangle. y 1 = 3.7x x 1.9 y 2 = x

36 Solving an Inequality Graphically We are interested in those values of x for which y 1 y 2. The solution are points for which the graph of y 2 (blue) lies on or above the graph of y 1 (red) You could pick a point to test, maybe the origin?? You try it.

37 E.g. 9 Solving an Inequality Graphically Solve the inequality x 3 5x 2 8 We write the inequality as: x 3 5x

38 E.g. 9 Solving an Inequality Graphically Then, we graph the equation y = x 3 5x in the viewing rectangle [ 6, 6] by [ 15, 15] The solution consists of those intervals on which the graph lies on or above the x-axis.