已知 $y = p \sin 2x + q$，其中 $p$ 和 $q$ 是正整数。$y$ 的最大值和最小值分别为 8 和 -2。

1. Formulate equations using the maximum and minimum values of the sine function

The maximum value of $\sin 2x$ is $1$ and the minimum value is $-1$.
Therefore, the maximum value of $y$ is $p(1) + q = 8$, so $p + q = 8$.
The minimum value of $y$ is $p(-1) + q = -2$, so $-p + q = -2$.

2. Solve the simultaneous equations to find $p$ and $q$

Adding the two equations:
$(p + q) + (-p + q) = 8 + (-2)$
$2q = 6 \implies q = 3$

Substituting $q = 3$ into $p + q = 8$:
$p + 3 = 8 \implies p = 5$

1. State the equation of the graph using the values of $p$ and $q$ from part (a)

From part (a), $p = 5$ and $q = 3$.
The equation is $y = 5 \sin 2x + 3$.

2. Determine the key features of the graph (period, amplitude, and principal axis)

Amplitude $= 5$
Principal axis is $y = 3$
Period $= \frac{360^\circ}{2} = 180^\circ$
Number of cycles in $360^\circ = 2$

3. Sketch the graph with correct shape, turning points, and intercepts

Draw a sine curve starting at $(0^\circ, 3)$.
First cycle:

• Maximum at $(45^\circ, 8)$
• Crosses axis at $(90^\circ, 3)$
• Minimum at $(135^\circ, -2)$
• Ends at $(180^\circ, 3)$
  Second cycle:
• Maximum at $(225^\circ, 8)$
• Crosses axis at $(270^\circ, 3)$
• Minimum at $(315^\circ, -2)$
• Ends at $(360^\circ, 3)$