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
y=x^3 + 6 x^2 + 14 x+ 7is the solid curve, and the approximate graph for
y= 3 x+ 1is the dashed line.
Are you confused?
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 calculat-
ing 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.
Here’s a challenge!
In the “challenge” at the end of Chap. 27, we solved the following two cubic functions as a two-by-two system:
y= 5 x^3 + 3 x^2 + 5 x+ 7and
y= 2 x^3 +x^2 + 2 x+ 5We 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.
Solution
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
Enter the Cubic 473