Final answer:
The equation for the soccer ball's path in ve rtex form is f(x) = -0.024(x - 25)² + 15, representing a parabolic trajectory with a vertex at (25, 15).
Step-by-step explanation:
To find the equation of the soccer ball's path in vertex form, we'll assume that the path follows a parabolic trajectory, which is typical for projectile motion. Vertex form for a parabola is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here, the vertex represents the highest point of the ball's trajectory, which is 15 feet high, making k = 15. The ball travels a horizontal distance of 50 feet before landing, which suggests that the vertex is at the midpoint of this distance, making h = 25 feet.
Assuming that the ball was kicked from the ground level and it lands back on the ground, we have two points: the starting point (0,0) and the landing point (50,0). We can plug the landing point into the vertex form to solve for 'a': 0 = a(50 - 25)² + 15, which simplifies to a = -15/625 = -0.024. Therefore, the completed vertex form is f(x) = -0.024(x - 25)² + 15.