The stream of water from a fountain follows a parabolic path. the stream reaches a maximum height of 7 feet, represented by a vertex of (4,7)
, and lands 8 feet from the water jet, represented by (8,0)
. write a function in vertex form that models the path of the stream.

First of all, I'm going to assume that we have a concave down parabola, because the stream of water is subjected to gravity.

If we need the vertex to be at , the equation will contain a term.

If we start with we have a parabola, concave down, with vertex at and a maximum of 0.

So, if we add 7, we will translate the function vertically up 7 units, so that the new maximum will be

We have

Now we only have to fix the fact that this parabola doesn't land at , because our parabola is too "narrow". We can work on that by multiplying the squared parenthesis by a certain coefficient: we want

such that:

Plugging these values gets us

As you can see in the attached figure, the parabola we get satisfies all the requests.

just take the area of the square and then the area of the circle then subtract the circle area from the squares area

step-by-step explanation: