Antoniomathgini

The following relationship gives the height, "h" in meters, of a soccer ball, as a function of the horizontal distance, d meters, the ball travels until it first hits the ground:

h = -0.025(d-20)^2 + 10

Q: If the origin were placed at the vertex of the parabola, what would be the equation of the curve?

I have given this some thought and figured that (h,k) of the vertex would double to make (2h,2k) since both the horizontal and vertical distances are increasing by 20 and 10.

This would make my new equation:

h = -0.025(d-40)^2 + 20

The answer provided is different. Apparently, the new curve would be:

h =-0.025(d^2)

Could you help me understand?

v8archie

1) Where is the vertex? $(d,h)=(20,10)$ can you see why?
2) If we translate the plane so that the $(d,h)=(20,10)$ becomes the origin, $(D,H)=(0,0)$, we are saying that $D=d-20$ and $H=h-10$.
3) The equation $h = -0.025(d-20)^2 + 10$ can then be transformed as follows:
\begin{align*}
h &= -0.025(d-20)^2 + 10 \\
h-10 &= -0.025(d-20)^2 \\
H &= -0.025D^2
\end{align*}