NECO 2022 · Paper 2 · Q11

(a)
Copy and complete the table of values below for the relation $y = 6 + 3x - 3x^2$ .

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

$y$ $-30$ $0$ $6$ $6$

Model answer

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

$y$ $-30$ $-12$ $0$ $6$ $6$ $0$ $-12$ $-30$

The new entries are $-12$ at $x = -2$ , $0$ at $x = 2$ , $-12$ at $x = 3$ and $-30$ at $x = 4$ . The row is symmetric about $x = \frac12$ , a useful check.
(b)
Draw the graph of the relation $y = 6 + 3x - 3x^2$ , using a scale of 2 cm to 1 unit on the $x$ -axis and 2 cm to 5 units on the $y$ -axis.
Model answer
Scale: 2 cm to 1 unit on the x-axis, 2 cm to 5 units on the y-axis.
Plot the eight points $(-3, -30)$ , $(-2, -12)$ , $(-1, 0)$ , $(0, 6)$ , $(1, 6)$ , $(2, 0)$ , $(3, -12)$ , $(4, -30)$ and join them with one smooth curve, not straight segments. The curve is symmetric about $x = 0.5$ and turns at $(0.5, 6.75)$ .
(c)
Using the same scales and axes, draw the graph of $y = 2 - x$ .
Model answer
Scale: 2 cm to 1 unit on the x-axis, 2 cm to 5 units on the y-axis.
Three points are enough for the line: $(-3, 5)$ , $(0, 2)$ and $(4, -2)$ . Draw it with a ruler across the whole grid.
(d)(i)
From your graph determine the maximum value of $y$ and the corresponding value of $x$ for which this occurs. (Enter $y_{max}$ and $x$ .)
Separate values with commas, e.g. 3, −2
(d)(ii)
From your graph determine the solution of the equation $6 + 3x - 3x^2 = 2 - x$ .
Separate values with commas, e.g. 3, −2

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$	$-30$		$0$	$6$	$6$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$y$	$-30$	$-12$	$0$	$6$	$6$	$0$	$-12$	$-30$

Worked solution (try it first)

Draw the axes to the given scales ( $x$ from $-3$ to $4$ , $y$ from $-30$ to $10$ ).
Plot the eight points $(-3, -30)$ , $(-2, -12)$ , $(-1, 0)$ , $(0, 6)$ , $(1, 6)$ , $(2, 0)$ , $(3, -12)$ , $(4, -30)$ and join them with a smooth curve.

For $y = 2 - x$ , plot $(-3, 5)$ , $(0, 2)$ and $(4, -2)$ and draw a straight line through them.

The top of the curve is halfway between the equal values at $x = 0$ and $x = 1$ , so it is at $x = 0.5$ .
There $y = 6 + 1.5 - 0.75 = 6.75$ .
The maximum value of $y$ is $6.75$ at $x = 0.5$ .

The solutions are the $x$ -values where the curve and the line cross: about $x = -0.7$ and $x = 2$ .
Check by algebra: $6 + 3x - 3x^2 = 2 - x$ gives $3x^2 - 4x - 4 = 0$ , which is $(3x + 2)(x - 2) = 0$ .
So $x = -\frac23$ or $x = 2$ .