We are given a parabola with vertex (-2,-4) that includes the point (1,-5), and we are asked to find the equation for the parabola.
The vertex form for a parabola is `y=a(x-h)^2+k` where the vertex is at (h,k) and a indicates whether the parabola opens up or down and how wide. (From the parent function `y=x^2` h performs a horizontal translation, k a vertical translation, and a a dilation.)
Here h=-2, k=-4 so it remains to find a:
Substitute the known values from the point (1,-5),h, and k into the equation and solve for a:
So the equation is `y=-1/9(x+2)^2-4` .
Scanning the list for an equivalent equation we note that b,d,and f cannot be the answer as h is wrong.
Answer A has a=-9, answer C has `a=-1/3` so these are wrong leaving only (e).
Rewrite `-1/9(x+2)^2-4` as `-(1/9(x+2)^2)-4=-(((x+2)^2)/9)-4`
`=-((x+2)/3)^2-4` which is answer (e).