Graphing with a Graphing Calculator

Transcription

1 APPENDIX C Graphing with a Graphing Calculator A graphing calculator is a powerful tool for graphing equations and functions. In this appendix we give general guidelines to follow and common pitfalls to avoid when graphing with a graphing calculator. See Appendix D for specific guidelines on graphing with the TI-83/84 graphing calculators. Selecting the Viewing Rectangle 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. 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 3a, b4 3 3c, d4 1x, y 0 a x b, c y d 6 as shown in Figure 1. We refer to this as the 3a, b4 by 3c, d4 viewing rectangle. (a, d) y=d (b, d) x=a x=b Figure 1 The viewing rectangle 3a, b4 by 3c, d4 (a, c) y=c (b, c) The graphing device draws the graph of an equation much as you would. It plots points of the form 1x, y for a certain number of values of x, equally spaced between a and b. 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. The machine connects each point to the preceding plotted point to form a representation of the graph of the equation. Example 1 Choosing an Appropriate Viewing Rectangle Graph the equation y x 3 in an appropriate viewing rectangle. SOLUTION Let s experiment with different viewing rectangles. We start with the viewing rectangle 3, 4 by 3, 4, so we set Xmin Ymin Xmax Ymax The resulting graph in Figure (a) (on the next page) is blank! This is because x 0, so x 3 3 for all x. Thus the graph lies entirely above the viewing rectangle, so this viewing rectangle is not appropriate. If we enlarge the viewing rectangle to 3 4, 44 by 3 4, 44, as in Figure (b), we begin to see a portion of the graph. C-1

2 C- APPENDIX C Graphing with a Graphing Calculator Now let s try the viewing rectangle 3 10, 104 by 3, 304. The graph in Figure (c) seems to give a more complete view of the graph. If we enlarge the viewing rectangle even further, as in Figure (d), the graph doesn t show clearly that the y-intercept is 3. So the viewing rectangle 3 10, 104 by 3, 304 gives an appropriate representation of the graph _4 4 _10 10 _0 0 _4 100 (a) (b) (c) (d) Figure Graphs of y x 3 Figure 3 Example Graphing a Cubic Equation Graph the equation y x 3 49x. SOLUTION Let s experiment with different viewing rectangles. If we start with the viewing rectangle 3, 4 by 3, 4 we get the graph in Figure 3. On most graphing calculators the screen appears to be blank, but it is not quite blank because the point 10, 0 has been plotted. It turns out that for all other x-values that the calculator chooses, the corresponding y-value is greater than or less than, so the resulting point on the graph lies outside the viewing rectangle. Let s use the zoom-out feature of a graphing calculator to change the viewing rectangle to the larger rectangle 3 10, 104 by 3 10, 104 In this case we get the graph shown in Figure 4(a), which appears to consist of vertical lines, but we know that cannot be true. If we look carefully while the graph is being drawn, we see that the graph leaves the screen and reappears during the graphing process. That indicates that we need to see more of the graph in the vertical direction, so we change the viewing rectangle to 3 10, 104 by 3 100,1004 The resulting graph is shown in Figure 4(b). It still doesn t reveal all the main features of the equation. It appears that we need to see still more in the vertical direction. So we try the viewing rectangle 3 10, 104 by 3 00, 004 The resulting graph is shown in Figure 4(c). Now we are more confident that we have arrived at an appropriate viewing rectangle. In Chapter 3, where third-degree

3 APPENDIX C Graphing with a Graphing Calculator C-3 polynomials are discussed, we learn that the graph shown in Figure 4(c) does indeed reveal all the main features of the equation _10 10 _10 10 _10 10 _10 _ (a) (b) (c) Figure 4 Graphing of y x 3 49x Interpreting the Screen Image Once a graph of an equation has been obtained by using a graphing calculator, we sometimes need to interpret what the graph means in terms of the equation. Certain limitations of the calculator can cause it to produce graphs that are inaccurate or need further modifications. Here are two examples. Example 3 Two Graphs on the Same Screen Graph the equations y 3x 6x 1 and y 0.3x. together in the viewing rectangle 3 1, 34 by 3., 1.4. Do the graphs intersect in this viewing rectangle? SOLUTION Figure (a) shows the essential features of both graphs. One is a pa rab ola, and the other is a line. It looks as if the graphs intersect near the point 11,. However, if we zoom in on the area around this point as shown in Figure (b), we see that although the graphs almost touch, they do not actually intersect. 1. _1.8 _1 3 Figure. (a) (b) You can see from Examples 1,, and 3 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.

4 C-4 APPENDIX C Graphing with a Graphing Calculator Example 4 Avoiding Extraneous Lines in Graphs Graph the equation y 1 1 x. SOLUTION Figure 6(a) shows the graph produced by a graphing calculator with viewing rectangle 3, 4 by 3, 4 In connecting successive points on the graph, the calculator produced a steep line segment from the top to the bottom of the screen. That line segment should not be part of the graph. The right side of the equation is not defined for x 1, so the calculator connects points on the graph to the left and right of x 1, and this produces the extraneous line segment. We can get rid of the extraneous near-vertical line by changing the graphing mode on the calculator. If we choose the Dot mode, in which points on the graph are not connected, we get the better graph in Figure 6(b). The graph in Figure 6(b) has gaps, so we have to interpret it as having the points connected but without creating the extraneous line segment. _ (a) Figure 6 Graphing y 1 1 x _ (b) Graphing Equations That Are Not Functions Most graphing calculators can only graph equations in which y is isolated on one side of the equal sign. Such equations are ones that represent functions (see page 16). The next example shows how to graph equations that don t have this property. Example Graphing a Circle Graph the circle x y 1. SOLUTION We first solve for y, to isolate it on one side of the equal sign. y 1 x Subtract x The graph in Figure 7(c) looks somewhat flattened. Most graphing calculators allow you to set the scales on the axes so that circles really look like circles. On the TI-83/84, from the ZOOM menu, choose ZSquare to set the scales appropriately. y 6"1 x Take square roots Therefore the circle is described by the graphs of two equations: y "1 x and y "1 x The first equation represents the top half of the circle (because y 0), and the second represents the bottom half of the circle (because y 0). If we graph the first equation in the viewing rectangle 3, 4 by 3, 4, we get the semicircle shown in Figure 7(a). The graph of the second equation is the semicircle in Figure 7(b).

5 APPENDIX C Graphing with a Graphing Calculator C- Graphing these semicircles together on the same viewing screen, we get the full circle in Figure 7(c). (a) (b) (c) Figure 7 Graphing the equation x y 1 C Exercises 1 6 Choosing a Window Use a graphing calculator or computer to decide which viewing rectangle (a) (d) produces the most appropriate graph of the equation. 1. y x 4 (a) 3, 4 by 3, 4 (b) 30, 44 by 30, 44 (c) 3 8, 84 by 3 4, 404 (d) 3 40, 404 by 3 80, y x 7x 6 (a) 3, 4 by 3, 4 (b) 30, 104 by 3 0, 1004 (c) 3 1, 84 by 3 0, 1004 (d) 3 10, 34 by 3 100, y 100 x (a) 3 4, 44 by 3 4, 44 (b) 3 10, 104 by 3 10, 104 (c) 3 1, 14 by 3 30, 1104 (d) 3 4, 44 by 3 30, y x 1000 (a) 3 10, 104 by 3 10, 104 (b) 3 10, 104 by 3 100, 1004 (c) 3 10, 104 by , (d) 3, 4 by 3 100, 004. y 10 x x 3 (a) 3 4, 44 by 3 4, 44 (b) 3 10, 104 by 3 10, 104 (c) 3 0, 04 by 3 100, 1004 (d) 3 100, 1004 by 3 00, y "8x x (a) 3 4, 44 by 3 4, 44 (b) 3, 4 by 30, 1004 (c) 3 10, 104 by 3 10, 404 (d) 3, 104 by 3, Graphing with a Graphing Calculator Determine an appropriate viewing rectangle for the equation, and use it to draw the graph. 7. y 100x 8. y 100x 9. y 4 6x x 10. y 0.3x 1.7x y " 4 6 x 1. y!1x y 0.01x 3 x 14. y x1x 6 1x 9 1. y 1 x x 16. y x x 17. y 1 0 x y x 0 x Intersection Points Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? 19. y 3x 6x 1, y "7 7 1 x ; 3 4, 44 by 3 1, y "49 x, y x ; 3 8, 84 by 3 1, y 6 4x x, y 3x 18; 3 6, 4 by 3, 04. y x 3 4x, y x ; 3 4, 44 by 3 1, Graph the circle x y 9 by solving for y and graphing two equations as in Example Graph the circle 1 y 1 x 1 by solving for y and graphing two equations as in Example 3.. Graph the equation 4x y 1 by solving for y and graphing two equations corresponding to the negative and positive square roots. (This graph is called an ellipse.) 6. Graph the equation y 9x 1 by solving for y and graphing the two equations corresponding to the positive and negative square roots. (This graph is called a hyperbola.)

6 ANSWERS to Appendix C Exercises 1. (c). (c) 3. (c) 4. (d). (c) 6. (d) _ No 0. No 1. Yes,. Yes, C-6