The vertex of this parabola is at (-2, -3). when the y-value is -2, the x-value is -5. what is the coefficient of the squared term in the parabola's equation?

Given:
vertex at (-2,-3), and a point on the parabola is (-5, -2)

The general formula for a parabola with vertex at (h,k) is given by
y=a(x-h)^2+k
in this problem, (h,k)=(-2,-3)
so
y=a(x+2)^2-3
We also know that one point on the parabola is (-5,-2), we will use this to solve for a.
-2=a(-5+2)^2-3
simplifying
-2=9a-3
solving for a
a=1/9
the complete equation of the parabola is therefore
y=(1/9)(x+2)^2-3

Check: at x=2, y=-3 => vertex
at x=-5, y=-2  => point on parabola.  so equation ok.