The table below shows the values of the relation \(y = 11 – 2x – 2x^{2}\) for \(-4 \leq x \leq 3\).
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | -13 | 11 |
(a) Copy and complete the table.
(b) Using a scale of 2 cm to 1 unit on the x- axis and 2 cm to 5 units on the y- axis, draw the graph of \(y = 11 – 2x – 2x^{2}\).
(c) Use your graph to find : (i) the roots of the equation \(11 – 2x – 2x^{2} = 0\) ; (ii) the values of x for which \(3 – 2x – 2x^{2} = 0\) ; (iii) the gradient of the curve at x = 1.
Explanation
(a)
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| \(x^{2}\) | 16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 |
| \(11\) | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 |
| \(-2x\) | 8 | 6 | 4 | 2 | 0 | -2 | -4 | -6 |
| \(-2x^{2}\) | -32 | -18 | -8 | -2 | 0 | -2 | -8 | -18 |
| y | -13 | -1 | 7 | 11 | 11 | 7 | -1 | -13 |
(b) 
(c)(i) x = -2.9 or 1.9
(ii) \(y + (3 - 2x - 2x^{2}) = 11 - 2x - 2x^{2}\)
\(y = 11 - 2x - 2x^{2} - 3 + 2x + 2x^{2}\)
\(y = 8\)
\(\therefore x = \text{-1.8 or 0.8}\)
(iii) \(y = 11 - 2x - 2x^{2}\)
\(\frac{\mathrm d y}{\mathrm d x} = -2 - 4x\)
Gradient at x = 1 : \(-2 - 4(1) = -2 - 4 = -6\)