(a) Copy and complete the following table of values for the relation (y = 2x^{2} – 7x – 3).
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 19 | -3 | -9 |
(b) Using 2 cm to 1 unit on the x- axis and 2 cm to 5 units on the y- axis, draw the graph of (y = 2x^{2} – 7x – 3) for (-2 leq x leq 5).
(c) From your graph, find the : (i) minimum value of y ;
(ii) gradient of the curve at x = 1.
(d) By drawing a suitable straight line, find the values of x for which (2x^{2} – 7x – 5 = x + 4).
Explanation
(a) (y = 2x^{2} – 7x – 3)
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| (2x^{2}) | 8 | 2 | 0 | 2 | 8 | 18 | 32 | 50 |
| (-7x) | 14 | 7 | 0 | -7 | -14 | -21 | -28 | -35 |
| (-3) | -3 | -3 | -3 | -3 | -3 | -3 | -3 | -3 |
| (y) | 19 | 6 | -3 | -8 | -9 | -6 | 1 | 12 |
(b)
(c)(i) Minimum value of y = -9
(ii) (y = 2x^{2} – 7x – 3)
Gradient = (frac{mathrm d y}{mathrm d x} = 4x – 7)
Gradient at x = 1 : (4(1) – 7 = 4 – 7 = -3).
(d) (2x^{2} – 7x – 5 = x + 4)
(2x^{2} – 7x – 5 + 2 = x + 4 + 2)
(2x^{2} – 7x – 3 = x + 6)
(implies y = x + 6)
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
| y | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
From the graph, x = -0.9 or 5.