Parabola Question?

Question

Assume all positions refer to the center of the ball.

Dwyane Wade shoots a lot of free throws. When he lets it go, it's 15 feet from the basket, and 7 feet off the ground. When the ball reaches its maximum height, it is 13 feet high. When the ball goes through the hoop, it is zero feet from the basket and 10 feet high.

Set this problem up so that the y-axis is a line through the center of the basket, and the x-axis is a line on the floor from under the center of the basket to directly under Mr. Wade.

Find the equation of the parabola that matches the ball's trajectory.

Best answer will explain the process.

Answer 1

Your coordinate system will have points at (-15,7) , (x1,13) and ((0,10) representing the position of the basketball at launch, position at hihest point, and position as it passes through the hoop, respectively.

The equation of the parabola is ax^2 +bx + c= 0. When x = 0, y= 10 so c = 10. So equation becomes ax^2+bx +10 = 0.

When x = -15, y= 7 so 225a -15b +10 = 7, or
225a - 15b = -3 <-- Eq 1
The axis of symmetry occurs when x1= -b/2a, so:
13 = a(-b/2a)^2 +b(-b/2a) +10
13= b^2/4a -b^2/2a +10
3= -b^2/4a
12a = -b^2
a = -b^2/12 <-- Eq 2
225(-b^2/12)- 15b = -3 {Sub Eq 1 into Eq 2}
-18.75b^2 -15b +3 = 0
b = [15+/- 18sqrt(2)]/-37.5
b = -.4 +/- .48sqrt(2)

Now you can solve for a

Something seems wrong here since there are two possible answers for b and both are irrational.

Answer 2

From the first sentence, you know that the point (x,y) = (15,7) is a point on the curve. The last sentence tells you that (0,10) is also a point on the curve.

If the function is f(x) = ax^2 + bx +c, the second sentence tells you that the y coordinate of the vertex is 13. But the vertex occurs when the derivative vanishes, or when f'(x) = 2ax + b = 0. That happens when x = -b/(2a), so f(-b/2a) = 13.

Using f(0) = 10, you find c = 10. From f(15) = 7, and f(-b/2a) = 13, you get two equations in the two unknowns a and b. Solve, and then you know f(x) completely.