Find the Coefficient of a Quadratic Function

Consider the quadratic function given by the equation:

$$ f(x) = ax^2 + bx + c $$

where $$ a $$, $$ b $$, and $$ c $$ are real constants. Suppose that the vertex of the parabola defined by this function is located at the point (3, -4). Additionally, you know that the function passes through the point (1, 0). Using this information, find the value of $$ a $$.