Question

A basketball free throw can be modeled by a parabolic path, which can be represented by the function b(x) = - 0.25 x^2 + 4x, where x is time, in seconds, and b(x) is the height of the ball off the ground in feet.

What is the VERTEX and what do the coordinates mean in the context of the problem?

please solve algebraically

Answer 1

The vertex of this problem is (8,16). This means that the highest height the ball reaches is at t = 8s, and this maximum height is of 16 feet.

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), y_(v))`

In which

`x_(v) = -(b)/(2a)`

`y_(v) = -(\Delta)/(4a)`

Where

`\Delta = b^2-4ac`

If a<0, the vertex is a maximum point, that is, the maximum value happens at
`x_(v)`, and it's value is
`y_(v)`.

b(x) = - 0.25 x^2 + 4x

Quadratic equation with
`a = -0.25, b = 4`.

Vertex:

`x_(v) = -(4)/(2(-0.25)) = 8`

`\Delta = 4^2-4(-0.25)(0) = 16`

`y_(v) = -(16)/(4(-0.25)) = 16`

The vertex of this problem is (8,16). This means that the highest height the ball reaches is at t = 8s, and this maximum height is of 16 feet.