TITLE: What are the coordinates of red point?
QUESTION [6 upvotes]: The function  $f (x)=-x^2+4$ "in red" is moving along the line $y=x+4$  " in black "  from green point  to  black point  and becomes in the place of blue graph  as shown in the following  graph


What are the coordinates  of red point?
I got the green point $(0,4)$ by using the equation  $y=x+4$ ,  and got the intersection  point of the two  parabolas $(2,0)$  and I stopped here.

REPLY [6 votes]: The other answers are good but here's another way to do it. Given a parabola with it's vertex on the line $y=x+4$ and $a=-1$, and a zero at $x=2$, the height of the parabola will be $x+4$, and $x$-distance from the vertex to the zero will be $|x-2|$. Thus, $(x+4)-|x-2|^2 = 0$, or $x^2-5x = 0$, which means that $x=0$ (the red parabola), or $x=5$ (the blue parabola). Since the distance from the vertex to each of it's roots are the same, we calculate $|x-2|$, and we can add and subtract that to $x$, the $x$-coordinate of the vertex. This gives us the two zeros of the blue parabola.