WAEC 2021 · Paper 2 · Q7

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

$x$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$

$y$ $19$ $-2$ $26$

Model answer

$x$ −4 −3 −2 −1 0 1 2 3 4

$y$ 34 19 8 1 −2 −1 4 13 26

For example, at $x = -4$ : $y = 2(16) + 4 - 2 = 34$ .
(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 $y = 2x^2 - x - 2$ for $-4 \le x \le 4$ .
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points). Its lowest point is about $(0.25, -2.1)$ .

For (d): (i) $2x^2 - 3x - 5 = 0$ is the same as $2x^2 - x - 2 = 2x + 3$ , so the roots are where the line meets the curve: $x = -1$ and $x = 2.5$ . (ii) $2x^2 - x - 2 < 0$ where the curve is below the $x$ -axis: $−0.8 < x < 1.3$ .
(c)
On the same axes, draw the graph of $y = 2x + 3$ .
Model answer
The straight line $y = 2x + 3$ goes through $(0, 3)$ and $(2, 7)$ : plot these (and one more, e.g. $(-2, -1)$ , as a check) and rule a line through them across the whole graph. See the model answer for (b), where it is drawn on the same axes.
(d)
Use the graph to find the: (i) roots of the equation $2x^2 - 3x - 5 = 0$ ; (ii) range of values of $x$ for which $2x^2 - x - 2 < 0$ .
Separate values with commas, e.g. 3, −2

$x$	−4	−3	−2	−1	0	1	2	3	4
$y$	34	19	8	1	−2	−1	4	13	26

The curve meets the line at the roots of 2x² − 3x − 5 = 0.

Worked solution (try it first)

$y = 2x + 3$ is a straight line: plot two or three points, for example $(-2, -1)$ , $(0, 3)$ and $(3, 9)$ , and join them with a ruler.

Rearrange: $2x^2 - 3x - 5 = 0$ is the same as $2x^2 - x - 2 = 2x + 3$ .
So its roots are the $x$ -values where the curve meets the line: $x = -1$ and $x = 2.5$ .

$2x^2 - x - 2 < 0$ where the curve is below the $x$ -axis, between the points where it crosses the axis: about $-0.8 < x < 1.3$ .