Interpreting Quadratic Functions in Vertex Form

We want to write the vertex form of the parabola whose vertex is $(\text{-}1,\text{-}1)$ and passes through point $(1,3).$ 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 $(\text{-}1,\text{-}1),$ we have that $h=\text{-}1$ and $k=\text{-}1.$ We can use these values to partially write our equation. $$\begin{array}{c}y=a(x-(\text{-}1){)}^2+(\text{-}1)\iff y=a(x+1{)}^2-1\end{array}$$ Finally, to find the value of $a,$ we will use the fact that the function passes through $(1,3).$ We can substitute $1$ for $x$ and $3$ for $y$ in the above equation and solve for $a.$ Now that we know that $a=1,$ we can complete the equation of the function. $$\begin{array}{c}y=(x+1{)}^2-1\end{array}$$