WAEC 2023 · Paper 2 · Q8

(a)
Copy and complete the table of values for $y = 2x^2 - x - 4$ for $-3 \le x \le 3$ .

$x$ −3 −2 −1 0 1 2 3

$y$ 17 −4

Model answer

$x$ −3 −2 −1 0 1 2 3

$y$ 17 6 −1 −4 −3 2 11

For example, at $x = -2$ : $y = 2(4) + 2 - 4 = 6$ .
(b)
Using a scale of 2 cm to 1 unit on the $x$ -axis and 2 cm to 2 units on the $y$ -axis, draw the graph of $y = 2x^2 - x - 4$ 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). Scale: 2 cm to 1 unit across, 2 cm to 2 units up.

For (c): (i) roots $x \approx −1.2 and 1.7$ ; (ii) $y$ increases as $x$ increases to the right of the lowest point, $x > 0.25$ ; (iii) the minimum point is $(0.25, -4.125)$ .
(c)
Use the graph to find the: (i) roots of the equation $2x^2 - x - 4 = 0$ ; (ii) values of $x$ for which $y$ increases as $x$ increases; (iii) minimum point of $y$ . Enter the two roots for (i).
Separate values with commas, e.g. 3, −2

$x$	−3	−2	−1	0	1	2	3
$y$	17	6	−1	−4	−3	2	11

Plot the curves, move them, and read values off the graph.

Worked solution (try it first)

With 2 cm to 1 unit across and 2 cm to 2 units up, plot the seven points and join them with a smooth U-shaped curve.

The roots are where the curve crosses the $x$ -axis: $x \approx -1.2$ and $x \approx 1.7$ .
(Exactly, $\frac{1 \pm \sqrt{33}}{4}$ , about $-1.19$ and $1.69$ .)

The lowest point of the curve is halfway between the roots, at $x = 0.25$ .
To its right the curve rises, so $y$ increases as $x$ increases for $x > 0.25$ (on this graph, $0.25 < x \le 3$ ).