SECTION 1.1 Graphing Utilities; Introduction to Graphing Equations 5 Step 2 Plot the points listed in the table as shown in Figure 9(a). Now connect the points to obtain the graph of the equation (a line), as shown in Figure 9(b). The graphs of the equations shown in Figures 9(b) and 10 do not show all the points that are on the graph. For example, in Figure 9(b), the point ( ) − 20, 37 is on the graph of y x2 3, = − + but it is not shown. Since the graph of y x2 3 = − + could be extended out as far as we please, we use arrows to indicate that the pattern shown continues. When constructing a graph, it is important to present enough of the graph so that any viewer of the illustration will “see” the rest of it as an obvious continuation of what is shown. This is referred to as a complete graph. One way to obtain a complete graph of an equation is to continue plotting points on the graph until a pattern becomes evident. Then these points are connected with a smooth curve following the suggested pattern. But how many points are sufficient? Sometimes knowledge about the equation tells us. For example, we will learn in How to Graph an Equation by Plotting Points Graph the equation: y x2 3 = − + Step-by-Step Solution Step 1 Find some points x y , ( ) that satisfy the equation. To find these points, choose values of x and use the equation to find the corresponding values for y. See Table 1. Graphing an Equation by Plotting Points Graph the equation: = y x2 EXAMPLE 3 Solution Table 2 provides several points on the graph. Plotting these points and connecting them with a smooth curve gives the graph (a parabola) shown in Figure 10. Figure 9 =− + y x2 3 x y –4 –2 2 4 6 8 –2 2 4 x y –4 –2 2 6 8 –2 2 4 (–2, 7) (2, –1) (–1, 5) (0, 3) (1, 1) (a) (b) (–2, 7) (2, –1) (–1, 5) (0, 3) (1, 1) x y x2 3 = − + x y, ( ) −2 ( ) − ⋅ − + = 2 2 3 7 ( ) −2, 7 −1 ( ) − ⋅ − + = 2 1 3 5 ( ) −1, 5 0 − ⋅ + = 2 0 3 3 ( ) 0, 3 1 − ⋅ + = 2 1 3 1 ( ) 1, 1 2 − ⋅ + = − 2 2 3 1 ( ) − 2, 1 Table 1 x y x2 = x y, ( ) −4 16 ( ) −4, 16 −3 9 ( ) −3, 9 −2 4 ( ) −2, 4 −1 1 ( ) −1, 1 0 0 ( ) 0, 0 1 1 ( ) 1, 1 2 4 ( ) 2, 4 3 9 ( ) 3, 9 4 16 ( ) 4, 16 Table 2 Figure 10 = y x2 x y –4 4 (4, 16) (–4, 16) (3, 9) (–3, 9) (2, 4) (–2, 4) (1, 1) (0, 0) (–1, 1) 5 10 15 20 EXAMPLE 2
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