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We want to write the vertex form of the parabola whose vertex is $(-1,-1)$ and passes through point $(1,3).$ To do so, let's first recall the vertex form of a quadratic function.
$y=a(x−h)_{2}+k $
In this form, $(h,k)$ is the vertex of the parabola. Since we are given that the vertex of our function is $(-1,-1),$ we have that $h=-1$ and $k=-1.$ We can use these values to partially write our equation.
$y=a(x−(-1))_{2}+(-1)⇔y=a(x+1)_{2}−1 $
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.
$y=(x+1)_{2}−1 $

$y=a(x+1)_{2}−1$

$3=a(1+1)_{2}−1$

Solve for $a$

AddTermsAdd terms

$3=a⋅2_{2}−1$

CalcPowCalculate power

$3=a⋅4−1$

AddEqn$LHS+2=RHS+2$

$4=4a$

DivEqn$LHS/4=RHS/4$

$1=a$

RearrangeEqnRearrange equation

$a=1$