A-level · Algebra & functions

The modulus functionExam-style questions

The modulus as a distance, and what follows from that. The graph of y = |x − a| and how to sketch y = |f(x)|; modulus equations, with two cases or by squaring; modulus inequalities, one stretch or two; why a solution must be checked when the other side contains x; and counting solutions with a horizontal line across the V.

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37 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
Solve $|x + 1| = 2x - 1$ .
Level 5 · 3 marksTry it
2
Figure 1Figure 1 shows a sketch of the graph $C_1$ with equation

$y = 3 - |2x + 1|$
(a) Write down

(i) the coordinates of the maximum point of $C_1$

(ii) an equation of the line of symmetry of $C_1$

(2 marks)
(b) Find the coordinates of the points where $C_1$ meets the coordinate axes.

(2 marks)
(c) On a copy of Figure 1, sketch the curve $C_2$ with equation

$y = 4 - x - \dfrac{1}{2}x^2$

Show clearly the coordinates of the points where $C_2$ meets the coordinate axes.

(3 marks)
(d) Using algebra, find the exact coordinates of the points where $C_1$ and $C_2$ intersect.

(Solutions relying entirely on calculator technology are not acceptable.)

(5 marks)
Level 5 · 12 marksTry it
3
Solve $|2x - 3| = x + 1$ .
Level 5 · 3 marksTry it
4
The diagram shows part of the graph of $y = f(x)$ , where $f(x) = 2|x - 1| + 3$ , $x \ge 0$ .(a) State the range of $f(x)$ .
(b) Solve the equation $f(x) = x + 4$ .
(c) Given that the equation $f(x) = k$ , where $k$ is a constant, has two distinct roots, state the set of possible values of $k$ .
Level 5 · 6 marksTry it
5
The graph of $y = |2x - 8|$ is shown in the diagram.(a) Write down the coordinates of the points where the graph meets the coordinate axes.

(2 marks)

The straight line $l$ has gradient $m$ and passes through the point $P(-1, -2)$ . The line $l$ meets the graph of $y = |2x - 8|$ at two distinct points.
(b) (i) Determine the set of possible values of $m$ .

(ii) Determine, in terms of $m$ , the $x$ -coordinates of these two points.

(7 marks)
Level 5 · 9 marksTry it

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