求 $3\sin\sin 2x - 2$ 的振幅和周期。 画出曲线 $y = 3\sin\sin 2x - 2$ 在 $0^\circ \le x \le 360^\circ$ 上的草图。

1. State the amplitude

For $y = 3\sin 2x - 2$, the amplitude is the magnitude of the coefficient of $\sin 2x$.
$\text{Amplitude} = 3$

2. Find the period

For a sine function $a\sin(bx)+c$, the period is $\dfrac{360^\circ}{b}$ with $b=2$.
$\text{Period} = \frac{360^\circ}{2} = 180^\circ$

1. Identify key values for the curve

From part (a): amplitude $=3$, period $=\dfrac{360^\circ}{2}=180^\circ$.

Vertical shift $-2$ gives the centre line $y=-2$.

Maximum value $=-2+3=1$, minimum value $=-2-3=-5$.

Number of complete cycles in $0^\circ\le x\le360^\circ = \dfrac{360^\circ}{180^\circ}=2$.

2. Plot key points and draw the curve

Mark the key points of one period and repeat:

• $x=0^\circ:\ y=-2$
• $x=45^\circ:\ y=1$ (maximum)
• $x=90^\circ:\ y=-2$
• $x=135^\circ:\ y=-5$ (minimum)
• $x=180^\circ:\ y=-2$

Repeat the same shape over $180^\circ\le x\le360^\circ$ (max at $225^\circ$, min at $315^\circ$).

Join with a smooth sinusoidal curve, labelling the axes and the max/min values $1$ and $-5$.