Use the following information to answer the next question
A parabola, $f(x)$, has a vertex at $(3,14)$ and a y-intercept at $(0,19)$.
Numeric Response
3. For the vertex form of its equation, $f(x)=a(x-p)^{2}+q$, the value of " $a$ ", to the nearest hundredth, is

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 \(\boxed{0.56}\).