Modelling the equation of a parabola.

Two parabolas are joined smoothly at the point B(30,20).

a) The parabola through A and B has a zero gradient at A(0,60). Find the equation of this parabola expressing all coefficients as exact values.

The general equation of a quadratic is:

$\displaystyle y = ax^2 + bx + c = 0$

$\displaystyle a(30)^2 + b(30) + c = 20$

$\displaystyle 900a + 30b + c = 20$

$\displaystyle a(0)^2 + b(0) + c = 60$

$\displaystyle c = 60$

$\displaystyle \frac{dy}{dx} = 2ax + b = 0$

$\displaystyle = 2a(0) + b = 0$

$\displaystyle b = 0$

$\displaystyle 900a + 30b + c = 20$

$\displaystyle 900a + 30(0) + 60 = 20$

$\displaystyle 900a = -40$

$\displaystyle a = \frac{-2}{45}$

Hence the equation of the parabola is:

$\displaystyle y = \frac{-2}{45}x^2 + 60 = 0$

Is this right so far? Can anyone confirm? Anybody got any ideas on how I'd go about answering b and c?

b) The parabola through B and C has a zero gradient at C. If the equation of this parabola is $\displaystyle y=a(x-p)^2$, find the exact values for a and p.

c) Write down the coordinates of this point.