Question 4.10b: Plot the graphs of the following: (a) y = 0.5x³ − 5x² + 8.5x......
Plot the graphs of the following:
(a) y = 0.5x³ − 5x² + 8.5x + 27 (b) y=−0.5x³ − 5x² + 8.5x + 27.
From the graphs estimate:
(i) The roots.
(ii) The turning points. Compare the graphs of (a) and (b).
First set up a table of values from which the graphs may be plotted. (See Table 4.8 and Figures 4.12a and 4.12b.) The roots and the turning points are summarised as follows:
(a) has only one root, x = –1.56 approximately
(a) has two turning points at x = 1.0, y = 31 and at x = 5.5, y = 5.6
(b) has three roots, at x = –11, x = –1.7 and x = 2.8 approximately
(a) has two turning points at x = –7, y = –107 and at x = 0.8, y = 30
Note: From an economic perspective, only the first quadrant (x > 0, y > 0) in Figures 4.12a and 4.12b is of interest. However, for a better understanding of mathematics, it is necessary to analyse the characteristics of the graphs in the other quadrants.
| Table 4.8 Points for plotting graphs in Worked Example 4.10b | ||
| x | Graph (a) | Graph (b) |
| -12 | -1659 | 69 |
| -8 | -617 | -105 |
| -4 | -119 | -55 |
| 0 | 27 | 27 |
| 4 | 13 | -51 |
| 8 | 31 | -481 |
| 2 | 237 | -1455 |
