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).

Step-by-Step

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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 |

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