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0606 P12 - Mar 2025 - Q2 - 3 marks
7184
The diagram shows the curve \(y=a\cos bx+c\) for \(-180^\circ\leqslant x\leqslant180^\circ\).
It is given that \(a\), \(b\) and \(c\) are integers.
Find the values of \(a\), \(b\) and \(c\).
Solution
Answer: \(a=5\), \(b=3\), \(c=-2\).
Start by taking the key information from the graph or diagram, such as intercepts, turning points, amplitudes, periods, or intersection points. Then use the relevant algebra to justify the result.
From the graph, the maximum value is \(3\) and the minimum value is \(-7\).
The midline is therefore
\(\frac{3+(-7)}{2}=-2,\)
so \(c=-2\).
The amplitude is
\(\frac{3-(-7)}{2}=5,\)
so \(a=5\).
There is a maximum at \(x=0^\circ\), and the next maximum occurs after \(120^\circ\). Therefore the period is \(120^\circ\).
For \(\cos bx\), the period is \(\frac{360^\circ}{b}\), so
\(\frac{360}{b}=120.\)
Hence \(b=3\).
The result has been obtained using the exact condition from the question, so no additional solutions are introduced.
