(a) Copy and complete the table of values for the relation y = 4x\(^2\) – 8x – 21, for -2.0 \(\leq\) x \(\leq\) 4.0
| x | -2.0 | -1.5 | -1.0 | 0.5 | 0.0 | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
| y | 11 | -9 | -21 | -24 | -21 | -9 | 0 |
(b) Using a scale of 2cm to 1 unit on the x-axis and 2cm to 5 units on the y-axis, draw the graph for the relation y = 4x\(^2\) – 8x – 21
(ii) Use the graph to find the solution set of
(\(\alpha\)) 4x\(^2\) – 8x = 3;
(\(\beta\)) 4x\(^2\) – 7x – 21 = 0
Explanation
(a) They completed the table of values as shown
| x | -2.0 | -1.5 | -1.0 | 0.5 | 0.0 | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
| y | 11 | 0 | -9 | -16 | -21 | -24 | -25 | -24 | -21 | -16 | -9 | 0 | 11 |
(b)(i) The graph
(b)(ii)(\(\alpha\), deduce that 4x\(^2\) - 8x - 21 = 3 - 21 so that y = -18. From the graph, the point of intersection of y = -18 and y = 4x\(^2\) - 8x - 21 was atx = {-0.35 \(\pm\) 0.1, 2.3 \(\pm\) 0.1}
(b)(ii)(\(\beta\)), the equation 4x\(^2\) - 8x - 21 = 0 which can be written as 4x\(^2\) - x - 7x - 21 = -x. This implies that 4x\(^2\) - 8x - 21 = -x.
But, y = 4x\(^2\) - 8x - 21 so that y = -x, Next was to draw the line y = -x and from the graph, the point of intersection with the curve y = 4x\(^2\) - 8x - 21 was at x = {3.3 \(\pm\) 0.1, -1.6 \(\pm\) 0.1}