上图展示了一只海豚跳出水面以看清海面的运动路径。该路径由一个二次函数 $y = ax^2 + bx + c$ 进行建模，如图所示。路径的尺寸如下表所示。

海豚能跳出水面多高？	450 cm
海豚能跃出水面多远？	600 cm

1. Identify the maximum height (y-coordinate)

The maximum height is given as $450\text{ cm}$.
So, $y = 450$.

2. Identify the x-coordinate of the highest point using symmetry

The total horizontal distance is $600\text{ cm}$.
Since the parabola starts at $(0,0)$ and ends at $(600,0)$, by symmetry, the highest point occurs at the midpoint of the x-intercepts.
$x = \frac{0 + 600}{2} = 300$

1. Set up the quadratic equation using the $x$-intercepts

The dolphin starts at the origin $(0,0)$ and re-enters the water at $x = 600$. Thus, the $x$-intercepts are $0$ and $600$.
The equation can be written in the factored form:
$y = ax(x - 600)$

2. Substitute the coordinates of the highest point to find $a$

From part (i), the highest point is $(300, 450)$. Substitute $x = 300$ and $y = 450$ into the equation:
$450 = a(300)(300 - 600)$
$450 = a(300)(-300)$
$450 = -90000a$
$a = -\frac{450}{90000} = -\frac{1}{200}$

3. Expand the equation to the required form $y = ax^2 + bx + c$

$y = -\frac{1}{200}x(x - 600)$
$y = -\frac{1}{200}x^2 + 3x$
(Here $a = -\frac{1}{200}$, $b = 3$, $c = 0$)