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.

step-by-step explanation:

y = | -2(-2) |     →     y = | 4 |     →     y = 4       ⇒   (-2, 4)

y = | -2(-1) |     →     y = | 2 |     →       y =   2       ⇒   (-1, 2)    

y = | -2(0) |     →     y = | 0 |     →       y = 0         ⇒   (0, 0)

y = | -2(1) |   c →     y = | -2 |     →     y =   2         ⇒   (1, 2)    

y = | -2(2) |     →     y = | - 4 |     →     y =   4         ⇒   (2, 4)