Identify the Quadratic Function: Parabola with Maximum Point (4,0)

To solve this problem, we'll use the vertex form of a parabolic function.

Recall that the vertex form of a parabola is given by:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola. In this problem, we are given a vertex at $(4, 0)$.

Step 1: Identify the vertex coordinates:

1. $h = 4$
2. $k = 0$

Step 2: Determine the sign of $a$.

We are informed that the point $(4, 0)$ is a maximum point, which means the parabola opens downward. For a downward-opening parabola, the coefficient $a$ must be negative.

Step 3: Substitute the identified values into the vertex form equation:

$y = a(x - 4)^2 + 0$

$y = a(x - 4)^2$

Since the parabola is downward-opening:

$a < 0$, for instance, $a = -1$

Thus, the equation is $y = -(x - 4)^2$.

This equation describes a parabola with a vertex at $(4, 0)$ and opens downward, achieving a maximum there. Therefore, the correct function corresponding to the parabola with a maximum point at $(4, 0)$ is:

$y=-(x-4)^2$