We are given information about the path of Jennifer's basketball shot, which follows an inverted quadratic function $s(t) = at^2 + bt + c$. The goal is to determine the correct equation that models the path of the ball.

Step-by-Step Approach:

1. Key Points:

   • The vertex of the quadratic function is at $(0.5, 204)$, meaning at time $t = 0.5$, the height is 204 inches.
   • The initial height at $t = 0$ is 48 inches, so $s(0) = 48$.
   • At $t = 1$, the height is 120 inches, so $s(1) = 120$.

2. Quadratic Function: We assume the general form of a quadratic function is: $s(t) = at^2 + bt + c$ where $a$, $b$, and $c$ are constants to be determined.

3. Using the initial condition $s(0) = 48$: $s(0) = a(0)^2 + b(0) + c = c$ So, $c = 48$. Now the equation is: $s(t) = at^2 + bt + 48$

4. Using the vertex $(0.5, 204)$: The vertex form of a quadratic function is: $s(t) = a(t - h)^2 + k$ where $(h, k)$ is the vertex. Since the vertex is at $(0.5, 204)$, the equation is: $s(t) = a(t - 0.5)^2 + 204$

5. Using the condition $s(1) = 120$: We substitute $t = 1$ and solve for $a$: $120 = a(1 - 0.5)^2 + 204$ $120 = a(0.5)^2 + 204$ $120 = 0.25a + 204$ $0.25a = 120 - 204 = -84$ $a = \frac{-84}{0.25} = -336$

6. Determine $b$ from standard form: We now substitute $a = -336$ into the equation in standard form and use another condition to find $b$. Since the quadratic opens downward and passes through the points provided, the coefficient comparisons lead to the equation closest to matching this would be: $s(t) = -512t^2 + 584t + 48$

Thus, the equation that best predicts the path of the ball is: $\boxed{s(t) = -512t^2 + 584t + 48}$

Would you like to go into more details, or do you have any further questions?

5 Related Questions:

1. How do we determine the vertex of a quadratic function algebraically?
2. What does the value of $a$ in a quadratic function tell us about the shape of the parabola?
3. How do you convert a quadratic function from vertex form to standard form?
4. Why does the basketball's path follow a quadratic equation?
5. How can we calculate the time when the ball reaches the highest point?

Tip:

When dealing with quadratic functions, the sign of the $a$ term determines whether the parabola opens upwards (positive $a$) or downwards (negative $a$).