图中显示了 $y = p\sin(qx) + r$ 图像的一部分。

1. Find the vertical shift $r$ using the maximum and minimum values

The maximum value is $3$ and the minimum value is $-1$.
$r = \frac{\text{max} + \text{min}}{2} = \frac{3 + (-1)}{2} = 1$

2. Find the amplitude $p$ using the maximum and minimum values

$p = \frac{\text{max} - \text{min}}{2} = \frac{3 - (-1)}{2} = 2$

3. Find the frequency multiplier $q$ using the period of the graph

From the graph, one full cycle is completed between $x = 0$ and $x = \pi$, so the period is $\pi$.
$\text{Period} = \frac{2\pi}{q}$
$\pi = \frac{2\pi}{q} \implies q = 2$

1. Identify the $x$-coordinate of the maximum point between $k$ and $h$

From part (i), the curve is $y = 2\sin(2x) + 1$. The first maximum occurs when $\sin(2x) = 1$, which gives $2x = \frac{\pi}{2} \implies x = \frac{\pi}{4}$.

2. Use the symmetry of the sine curve to relate $k$, $h$, and the maximum point

The points $(k, 2)$ and $(h, 2)$ have the same $y$-value and lie on either side of the first maximum. Due to the symmetry of the sine curve, the $x$-coordinate of the maximum is the midpoint of $k$ and $h$.
$\frac{k + h}{2} = \frac{\pi}{4}$
$k + h = \frac{\pi}{2}$