Step 1 :The vertex form of a parabola is given by \(f(x)=a(x-p)^{2}+q\), where \((p, q)\) is the vertex of the parabola. We know that the vertex is at \((3,14)\), so \(p=3\) and \(q=14\).
Step 2 :We also know that the parabola passes through the y-intercept at \((0,19)\). We can substitute these values into the equation to solve for \(a\).
Step 3 :Substituting \(p=3\), \(q=14\), and \((x,y)=(0,19)\) into the equation, we get \(19=a(0-3)^{2}+14\), which simplifies to \(19=9a+14\).
Step 4 :Solving for \(a\), we get \(a=\frac{5}{9}\).
Step 5 :Rounding to the nearest hundredth, we get \(a=0.56\).
Step 6 :Final Answer: The value of \(a\), to the nearest hundredth, is \(\boxed{0.56}\).