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Finally, we plot the rest We can fill in the graphs by plotting the remaining points in the table In Fig 28-3, the approximate graph for
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y = x 3 + 6x 2 + 14x + 7 is the solid curve, and the approximate graph for y = 3x + 1 is the dashed line
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Figure 28-3 doesn t show the relationship between the curve and the line very well in the vicinity of the solution points If you want to get a finer graph in that region, you can plot points at intervals of 1/2 unit, 1/5 unit, or even 1/10 unit for x-values between 4 and 0 or between 5 and 1 You can also include more points farther out, say for x-values of 7, 10, and 15 on the negative side and 5, 10, and 15 on the positive side A programmable calculator, or a personal computer with calculating software installed, makes an excellent assistant for this process, and can save you from having to do a lot of tedious arithmetic You might also find a site on the Internet that can calculate values of a linear, quadratic, cubic, or higher-degree function based on coefficients, the constant, and input values you choose
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In the challenge at the end of Chap 27, we solved the following two cubic functions as a two-by-two system: y = 5x 3 + 3x 2 + 5x + 7 and y = 2x 3 + x 2 + 2x + 5 We got one real solution, (x, y) = ( 2/3, 95/27), and two complex-conjugate solutions Draw a graph showing these two functions, along with the real solution point
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Table 28-4 shows several values of x, along with the resulting function values The solution is in the middle, written in bold The span of values for the input is from 3 to 2, while the span of values of the functions is from 116 to 69 Let s make each increment on the x axis represent 1/2 unit, and each increment on the y axis represent 10 units With six divisions going out from 0 to the left and six to the right, that gives us a span from 3 to 3 for x For y, we have eight divisions going up and 12 divisions going down, and that s a span
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474 More Two-by-Two Graphs
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Table 28-4 Selected values for graphing the functions y = 5x 3 + 3x 2 + 5x + 7 and y = 2x 3 + x 2 + 2x + 5 The bold entry indicates the real solution
x 3 2 1 2/3 Approx 067 0 1 2 5x 3 + 3x 2 + 5x + 7 116 31 0 95/27 Approx 352 7 20 69 2x 3 + x 2 + 2x + 5 46 11 2 95/27 Approx 352 5 10 29
of 120 to 80, more than enough to include all the function values in Table 28-4 To plot the solution point, we can convert the values to decimal form and go to a couple of decimal places Then we get (x, y) = ( 067, 352) This point is shown as a solid dot in Fig 28-4 Once we ve plotted it, we fill in the graphs of the functions The approximate graph for y = 5x 3 + 3x 2 + 5x + 7 is the solid curve, and the approximate graph for y = 2x 3 + x 2 + 2x + 5 is the dashed curve
Are you still confused
Do you wonder about the cubic curves in Figs 28-3 and 28-4 They re a lot different from the graphs of quadratics! The graph of a cubic function always has one of six characteristic shapes, as shown in Fig 28-5 They all look rather like distorted images of the letter S tipped on its side, perhaps flipped over backward, and then extended forever upward and down Unlike a quadratic function, which has a limited range with an absolute maximum or an absolute minimum, a cubic function always has a range that spans the entire set of real numbers, although it can have a local maximum and a local minimum The graph of a cubic function also has something else that you ll never see in the graph of a quadratic: an inflection point, where the curvature reverses direction The contour of the graph depends on the signs and values of the function s coefficients and constant If you want to get familiar with how the graphs of various cubic functions look, you can conjure up a few cubic functions with assorted coefficients and constants Then plot a couple of dozen points for each function, and connect the points with smooth curves But don t spend too much time at this A book devoted to the art of graphing cubic and higher-degree functions could consume thousands of pages! You ll learn more about graphing functions when you take a course in calculus
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