A Football is thrown by a quarterback to a receiver. When the quarterback throws it, the ball is at the height of 6 feet above the ground. When the receiver catches it, he is 30 yards away from the quarterback and the ball is again at a height of 6 feet above the ground. The maximum height the ball reaches, in its path from the quarterback to the receiver, is 12 feet above the ground
Find the quadratic function that models the trajectory of the football, as thrown from the quarterback to the receiver. In this model, let y denote the height of the ball, in feet, and let x denote the dis
distance, in yards, from the quarterback.
To find the quadratic function that models the trajectory of the football, we need to determine the equation of the parabolic path formed by the ball. This can be done by considering the three given points: (0, 6), (30, 6), and (15, 12).
Let's start by using the vertex form of a quadratic function, which is given by the equation: y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
Using the vertex form, we know that the vertex lies halfway between the x-coordinate values of the two given points (0 and 30). Therefore, the x-coordinate of the vertex is (0 + 30) / 2 = 15.
Additionally, we know the vertex lies at a height of 12 feet above the ground, so k = 12.
The vertex form equation now becomes: y = a(x-15)^2 + 12.
To find the value of 'a', we can substitute any of the three given points (0, 6), (30, 6), or (15, 12) into the equation.
Let's use the point (0, 6) to calculate 'a':
6 = a(0-15)^2 + 12
6 = a(225) + 12
6 = 225a + 12
6 - 12 = 225a
-6 = 225a
a = -6/225
a = -2/75
Hence, the quadratic function that models the trajectory of the football is:
y = (-2/75)(x-15)^2 + 12.