Using a scale of 2cm to 1 unit on the x- axis and 1cm to 1 unit on the y- axis, draw on the same axes the graphs of (y = 3 + 2x – x^{2}; y = 2x – 3) for (-3 leq x leq 4). Using your graph:
(i) solve the equation (6 – x^{2} = 0);
(ii) find the maximum value of (3 + 2x – x^{2});
(iii) find the range of x for which (3 + 2x – x^{2} leq 1), expressing all your answers correct to one decimal place.
Explanation
(y = 3 + 2x – x^{2}) and (y = 2x – 3)
Table of values for the equation for (-3 leq x leq 4)
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| (y = 3 + 2x – x^{2}) | -12 | -5 | 0 | 3 | 4 | 3 | 0 | 5 |
| (y = 2x – 3) | -9 | -7 | -5 | -3 | -1 | 1 | 3 | 5 |
(i) (6 – x^{2} = 0)
(6 – 3 + 2x – x^{2} = 2x – 3)
(3 + 2x – x^{2} = 2x – 3)
(therefore y = 2x – 3)
Read the point where the two equations intersect on the graph.
x = -2.6 and x = 2.5.
(ii) Maximum value of (3 + 2x – x^{2}) is at y = 4.
(iii) Range for which (3 + 2x – x^{2} leq 1) is represented by the shaded portion in the graph.