The answer
(a) min \(= -1\), max \(= 7\)
(b) sketch (amplitude \(4\), midline \(y = 3\), period \(180^{\circ}\))
O-Level A-Math 2021 Specimen Paper 1 Question 2 · Verified worked solution by the Genius Plus Academy teaching team
The question
\(y = 3 - 4\sin 2x\).
(a) State the minimum and maximum values of \(y\). [2]
(b) Sketch \(y = 3 - 4\sin 2x\) for \(0^{\circ} \leqslant x \leqslant 360^{\circ}\). [3]
(a) Since \(-1 \leqslant \sin 2x \leqslant 1\), \(y = 3 - 4\sin 2x\) ranges from \(3 - 4(1) = -1\) to \(3 - 4(-1) = 7\). So minimum \(y = -1\), maximum \(y = 7\).
(b) The graph oscillates about the midline \(y = 3\) with amplitude \(4\) and period \(\dfrac{360^{\circ}}{2} = 180^{\circ}\). Because of the \(-4\sin\), it starts at \((0, 3)\) falling to the minimum \(-1\) at \(x = 45^{\circ}\), rises through \((90^{\circ}, 3)\) to the maximum \(7\) at \(x = 135^{\circ}\), back to \((180^{\circ}, 3)\), then repeats the same cycle on \([180^{\circ}, 360^{\circ}]\) (minimum at \(225^{\circ}\), maximum at \(315^{\circ}\), ending at \((360^{\circ}, 3)\)).
Answer: (a) min \(= -1\), max \(= 7\)
(b) sketch (amplitude \(4\), midline \(y = 3\), period \(180^{\circ}\))
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
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It is a range of a+b kx question from Trig functions, identities & equations, worth 5 marks: 2 + 3.
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