SECTION 1.1 Graphing Utilities; Introduction to Graphing Equations 3 Graphing Utilities All graphing utilities (that is, all graphing calculators and all computer software graphing packages) graph equations by plotting points on a screen. The screen itself actually consists of small rectangles called pixels. The more pixels the screen has, the better the resolution. Most newer graphing calculators have at least 100 pixels per inch; most newer smartphones have at least 300 pixels per inch. When a point to be plotted lies inside a pixel, the pixel is turned on (lights up). The graph of an equation is a collection of pixels. Figure 4(a) shows how the graph of y x2 = looks on a TI-84 Plus CE graphing calculator, and Figure 4(b) shows the same graph using Desmos. The screen of a graphing utility displays the coordinate axes of a rectangular coordinate system. However, the scale must be set on each axis. The smallest and largest values of x and y to be included in the graph must also be set. This is called setting the viewing rectangle or viewing window. Figure 5 shows a typical viewing window on a TI-84 Plus CE. To set the viewing window, values must be given to the following expressions: Xmin: the smallest value of x Xmax: the largest value of x Xscl: the number of units per tick mark on the x-axis Ymin: the smallest value of y Ymax: the largest value of y Yscl: the number of units per tick mark on the y-axis Figure 6 illustrates these settings and their relation to the Cartesian coordinate system. Figure 4 (a) y x2 = on a TI-84 Plus CE (b) y x2 = using Desmos Figure 5 Viewing window on a TI-84 Plus CE Figure 6 Figure 7 4 24 23 3 If the scale used on each axis is known, the minimum and maximum values of x and y shown on the screen can be determined by counting the tick marks. Look again at Figure 5. For a scale of 1 on each axis, the minimum and maximum values of x are 10 − and 10, respectively; the minimum and maximum values of y are also 10 − and 10. If the scale is 2 on each axis, then the minimum and maximum values of x are 20 − and 20, respectively; and the minimum and maximum values of y are 20 − and 20, respectively. Conversely, if the minimum and maximum values of x and y are known, the scales can be determined by counting the tick marks displayed.This text follows the practice of showing the minimum and maximum values of x and y in illustrations so that the reader will know how the viewing window was set. See Figure 7. The numbers outside of the viewing window stand for = − = − = = = = X Y X Y X Y min 3, min 4, max 3, max 4, scl 1 scl 2 Now Work PROBLEMS 11 AND 21
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