To represent the path of the basketball, we can use a quadratic equation in the form of y = ax^2 + bx + c, where y represents the height of the basketball and x represents the horizontal distance from the starting point.
Given that Kim throws the basketball up from the ground towards the basketball hoop and it returns back to the gym floor 5 feet from the hoop, we can determine that the vertex of the parabolic path occurs at the midpoint between the starting point and the landing point. This midpoint can be calculated as (20 + 5) / 2 = 12.5 feet.
Since the highest point of the trajectory occurs at the vertex, the y-coordinate of the vertex represents the maximum height of the basketball. Since we are starting from the ground, the vertex will be the highest point, which means the maximum height will be the y-coordinate of the vertex.
Given that the maximum height is not provided, we cannot determine the exact value of the y-coordinate. Instead, we can represent it as 'h'. Therefore, the vertex of the parabolic path can be expressed as (12.5, h).
We can also determine that the basketball lands back on the ground at the starting point, which means the y-coordinate at this point is 0. Therefore, (20, 0) is also a point on the parabolic path.
Using these two points, we can find the equation of the parabolic path.
Using the vertex form equation:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabolic path, we can substitute the coordinates of the vertex (12.5, h) to get:
y = a(x - 12.5)^2 + h
Substituting the coordinates (20, 0) into the equation, we get:
0 = a(20 - 12.5)^2 + h
Simplifying this equation, we have:
a(7.5)^2 + h = 0
The equation to represent the path of the basketball is:
y = a(x - 12.5)^2 + (-a)(7.5)^2
where 'a' represents the coefficient determining the shape and direction of the parabola, 'x' represents the horizontal distance from the starting point, and 'y' represents the height of the basketball.