WAEC 2022 · Paper 2 · Q6

The graph shows the relation of the form $y = mx^2 + nx + r$ , where $m$ , $n$ and $r$ are constants. Using the graph:

Scale: 2 cm to 2 units on the x-axis and 2 cm to 10 units on the y-axis.

(a)
State the scale used on both axes.
Show the answer
$x$ : 2 cm to 2 units; $y$ : 2 cm to 10 units
(b)
Find the values of $m$ , $n$ and $r$ .
Separate values with commas, e.g. 3, −2
(c)
Find the gradient of the line through $P$ and $Q$ .
(d)
State the range of values of $x$ for which $y > 0$ .
Show the answer
$-2 < x < 4$

Try it on a graph

Drag-free check: does y = −x² + 2x + 8 really pass through P and Q? Edit the constants to see.

Worked solution (try it first)

(a)

On the $x$ -axis, 2 cm represents 2 units.
On the $y$ -axis, 2 cm represents 10 units.

(b)

The curve crosses the $x$ -axis at $x = -2$ and $x = 4$ , so $(x + 2)$ and $(x - 4)$ are factors.
It has a highest point, so the $x^2$ term is negative: $y = -(x + 2)(x - 4) = -x^2 + 2x + 8$ .
So $m = -1$ , $n = 2$ and $r = 8$ .
(Check: the curve crosses the $y$ -axis at $r = 8$ ✓.)

(c)

From the graph, $P(-5, -27)$ and $Q(3, 5)$ .
The rise is $5 - (-27) = 32$ and the run is $3 - (-5) = 8$ , so the gradient is $\frac{32}{8} = 4$ .

(d)

$y > 0$ where the curve is above the $x$ -axis, between the roots: $-2 < x < 4$ .

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