WAEC 2017 · Paper 2 · Q6

(a)
Copy and complete the table of values for the relation $y = 3x^2 - x - 11$ , for $-3 \le x \le 3$ .

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

$y$ $3$ $-7$

Model answer

$x$ −3 −2 −1 0 1 2 3

$y$ 19 3 −7 −11 −9 −1 13

For example, at $x = -3$ : $y = 3(9) + 3 - 11 = 19$ .
(b)
Using scales of 2 cm to 1 unit on the $x$ -axis and 2 cm to 5 units on the $y$ -axis, draw the graph of $y = 3x^2 - x - 11$ , for $-3 \le x \le 3$ .
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points).

For (c): (i) roots $x \approx −1.8 and 2.1$ ; (ii) the minimum value is about $-11.1$ (at $x \approx 0.2$ ); (iii) draw the tangent at $(1, -9)$ and use two points far apart on it: the gradient is about 5.
(c)
Use the graph to find the: (i) roots of $3x^2 - x - 11 = 0$ ; (ii) minimum value of $y$ ; (iii) gradient of the curve at the point $x = 1$ .
Separate values with commas, e.g. 3, −2

$x$	−3	−2	−1	0	1	2	3
$y$	19	3	−7	−11	−9	−1	13

The curve and its tangent at x = 1 (gradient 5).

Worked solution (try it first)

Substitute each $x$ into $y = 3x^2 - x - 11$ .
For example, $x = -3$ gives $27 + 3 - 11 = 19$ and $x = 3$ gives $27 - 3 - 11 = 13$ .
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

$y$ $19$ $3$ $-7$ $-11$ $-9$ $-1$ $13$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$19$	$3$	$-7$	$-11$	$-9$	$-1$	$13$

With 2 cm to 1 unit on the $x$ -axis and 2 cm to 5 units on the $y$ -axis, plot the seven points and join them with a smooth U-shaped curve.

The roots are where the curve crosses the $x$ -axis: $x \approx -1.8$ and $x \approx 2.1$ .

The lowest point is just right of $x = 0$ : the minimum value is $y \approx -11.1$ .

Draw the tangent at $(1, -9)$ .
It passes through about $(0, -14)$ and $(2, -4)$ , so the rise is $-4 - (-14) = 10$ and the run is $2 - 0 = 2$ .
Its gradient is $10 \div 2 = 5$ .
(Check: $\frac{dy}{dx} = 6x - 1 = 5$ at $x = 1$ .)