Parabola - how to write equation of parabola when co-ordinates are given..

Question

What will be the equation of parabola touching the $x$-axis at $(3,0)$ and and the $y$-axis at $(0,4)$.

Answer 1

If $y(x)=ax^2+bx+c$, then I assume, that you have to determine $a,b$ and $c$ under the conditions

$y(0)=4$, $y(3)=0$ and $y'(3)=0$.

I am right ?

Answer 2

I guess that you would like to know the equation of the parabola of the form $y=ax^2+bx+c$ whose only intersections with the coordinate axes are $(3,0)$ and $(0,4)$.

In this case note that the parabola is tangent to the $x$-axis at $(3,0)$ (why?). Therefore $y=a(x-3)^2$.

Can you take it from here?