A-level · Algebra & functions

Inequalities on graphs and regionsExam-style questions

Reading an inequality as a region of the plane. The boundary as the matching equation; which side of a curve or line to shade, and testing a point; dashed boundaries for strict inequalities and solid ones for inclusive; a region as the overlap of several inequalities; finding its corners by solving the boundaries together; and writing down the inequalities for a shaded region.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Figure 1

The curve $C$ in Figure 1 has equation $y = \text{f}(x)$ , where

$\text{f}(x) = 2x^2 - 8x + 6$
(a) Write $2x^2 - 8x + 6$ in the form $a(x + b)^2 + c$ , where $a$ , $b$ and $c$ are constants to be found.

(3 marks)

The curve $C$ has its minimum point at $V$ .
(b) Write down the coordinates of $V$ .

(1 mark)

The straight line $l$ passes through $V$ and through the point where $C$ crosses the $y$ -axis.
(c) Show that $l$ has equation $y = 6 - 4x$

(3 marks)

The region $R$ , shaded in Figure 1, has $C$ , $l$ and the $x$ -axis as its boundaries.
(d) Use inequalities to define the region $R$ .

(3 marks)
Level 5 · 10 marksTry it
2
Figure 1

The curve $C$ , sketched in Figure 1, has equation

$y = 14 + 12x - 2x^2$
(a) Write $14 + 12x - 2x^2$ in the form $p + q(x + r)^2$ , where $p$ , $q$ and $r$ are constants to be found.

(3 marks)

The maximum turning point of $C$ is $T$ .
(b) Deduce the coordinates of $T$

(2 marks)

The normal to $C$ at the point $D$ on $C$ is the line $l$ , which has equation $x + 4y = k$ , where $k$ is a constant.
(c) (i) Find the coordinates of $D$ .

(ii) Show that $k = 122$

(6 marks)

In Figure 1 the region $R$ , shown shaded, has boundaries $C$ , $l$ and the $y$ -axis.
(d) Use inequalities to define $R$ .

(3 marks)
Level 5 · 14 marksTry it