Interpreting Quadratic Functions in Vertex Form

We want to write the vertex form of the parabola whose vertex is $(1,9)$ and passes through point $P(4,12).$ To do so, let's first recall the vertex form of a quadratic function. $$\begin{array}{c}y=a(x-h{)}^2+k\end{array}$$ In this form, $(h,k)$ is the vertex of the parabola. Since we are given that the vertex of our function is $(1,9),$ we have that $h=1$ and $k=9.$ We can use these values to partially write our equation. $$\begin{array}{c}y=a(x-1{)}^2+9\end{array}$$ Finally, to find the value of $a,$ we will use the fact that the function passes through $(4,12).$ We can substitute $4$ for $x$ and $12$ for $y$ in the above equation and solve for $a.$ Now that we know that $a=\frac{1}{3},$ we can complete the equation of the function. $$\begin{array}{c}y=\frac{1}{3}(x-1{)}^2+9\iff y=\frac{1}{3}(x-1{)}^2+9\end{array}$$