There are only three pieces of information available:
- the graph passes through (0,0) and (6,0)
- the symmetry axis is $x$ = 3
- the graph is downward
My attempt:
I've tried to work on that problem and got quadratic function $y = ax^2 - 6ax$. However, I couldn't find the value of $a$ and of course the coordinate of the maximum value.
On
Since it is quadratic then you can represent it as $$ ax^2 + bx + c = 0 $$ and this must be true for both of your data points, i.e., $$ 0 = a(0)^2 + b(0) + c $$ and $$ 0 = a(6)^2+ b(6) + c $$ The first data point's equation leads to $c=0$. Then the second gives $ 36a + 6b = 0 $ or $6a+b=0$. Substituting $b=-6a$ gives you your equation $$ y = ax^2-6ax $$
With what's given, there isn't sufficient information to find a single value of $a$ that will give the minimum (it is $-9a$). If you had another data point it would give enough for a unique solution. https://www.desmos.com/calculator/da1iddugam
The quadratic functions which vanish at $x=0$ and $x=6$ are one-parameter family, and their equation, as you found it, is: $$y=ax(x-6).$$ There's a maximum only if $a<0$. If $a>0$, the function has a minimum. In any case the extremum is obtained at the midsum of the roots, i.e. at $x=3$. Hence the maximum, if $a<0$, is $$y(3)=-9a.$$