CAIE A-Level Further Mathematics 1.2.2 Related Graphs

Practise constructing modulus and other related graphs from a known curve, then using exact critical values, symmetry and branch behaviour to solve inequalities.

Exam points

transform the source curve while preserving intercepts and showing reflected branches or symmetry
solve both boundary equations exactly, then use the sketch to select the required inequality intervals

[Maximum number: 6]

The curve C has equation y=f(x), where $\mathrm{f}(x)=\frac{x^{2}+2}{x^{2}-x-2}$ .

(a)

Sketch the curve with equation $y=\frac{1}{\mathrm{f}(x)}$ .

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(b)

Find the set of values for which $\frac{1}{\mathrm{f}(x)}<\mathrm{f}(x)$ .

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$\begin{aligned} \frac{x^{2}+2}{x^{2}-x-2}=1 \text { or } \frac{x^{2}+2}{x^{2}-x-2}=-1 x+4=0 \quad \text { or } \quad 2 x^{2}-x=0 \end{aligned}$

M2

Finds critical points, award M1 for each case.

$x=-4 \quad$ or $x=0, \quad x=\frac{1}{2}$

A1

$-4<x<-1, \quad 0<x<\frac{1}{2}, x>2$

B1

Must have three distinct regions. Condone $\leqslant-1$
and $\geqslant 2$ .

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1.2.2—Related graphs question 1

Question (a)

Question (b)