Basketball Shot Parabola Simulation

Drag the sliders to change the shot parameters and click the buttons to start or reset the simulation.

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Math Exercise: Solving a Parabolic Equation

The trajectory of a basketball can be approximated as a quadratic function. Assume a shot's path follows the function $y = ax^2 + bx + c$. The shot is released from point $(0, 2)$, and the basketball's highest point is $(5, 8)$.

What is the function that describes the trajectory of this parabola?

Solution:

This problem is a classic example of determining a quadratic function based on points on its parabola.

  1. Use the Vertex Form
    The vertex form of a quadratic function is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
    The problem gives the highest point (the vertex) as $(5, 8)$. Substituting this into the vertex form gives: $$y = a(x - 5)^2 + 8$$
  2. Solve for $a$ using the initial point
    The parabola passes through the initial point of the shot, $(0, 2)$. Substitute these coordinates into the vertex form to solve for the coefficient $a$: $$2 = a(0 - 5)^2 + 8$$ $$2 = 25a + 8$$ $$-6 = 25a$$ $$a = -0.24$$
  3. Expand the function
    Substitute the value of $a$ back into the vertex form and expand it to get the standard form $y = ax^2 + bx + c$: $$y = -0.24(x - 5)^2 + 8$$ $$y = -0.24(x^2 - 10x + 25) + 8$$ $$y = -0.24x^2 + 2.4x - 6 + 8$$ $$y = -0.24x^2 + 2.4x + 2$$

Therefore, the correct function is **$y = -0.24x^2 + 2.4x + 2$**.

Math Exercise 2: Basketball Shot Analysis

The basketball follows a parabolic path from point (0, 2) to the hoop at (10.5, 3.05). Use the sliders to adjust the initial velocity and shot angle, then predict the outcome.

Question 1: If the initial velocity is 12 m/s and the shot angle is 50°, will the basketball hit the hoop?

Solution:

  1. Question 1:
    Using the sliders for 12 m/s at 50°, simulate the trajectory. Observing the graph, the basketball does not reach the hoop horizontally before descending, so the shot misses.
    Answer: B. No, it will miss the hoop.