The equation of the parabola given is \( y = -\frac{1}{24}x^2 + 2x \). To interpret key features of this parabola in the context of the baseball game, we will analyze the vertex and the intercepts.
Vertex
The vertex of a parabola expressed in the standard form \( y = ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
In this case:
- \( a = -\frac{1}{24} \)
- \( b = 2 \)
Calculating the x-coordinate of the vertex:
\[ x = -\frac{2}{2 \times -\frac{1}{24}} = -\frac{2}{-\frac{1}{12}} = 24 \]
To find the corresponding y-coordinate, substitute \( x = 24 \) back into the equation:
\[ y = -\frac{1}{24}(24^2) + 2(24) \] \[ y = -\frac{1}{24}(576) + 48 \] \[ y = -24 + 48 = 24 \]
Thus, the vertex of the parabola is at the point \( (24, 24) \).
Interpretation
The vertex \( (24, 24) \) represents the highest point of the ball's path. In the context of the baseball game, this means that the ball reaches its maximum height of 24 units (likely feet) at a horizontal distance of 24 units from the point where it was hit. This signifies the peak of the ball's trajectory during its flight, right before it begins to descend back towards the field.
Intercepts
Next, we can also find the x-intercepts (where the ball hits the ground) by setting \( y = 0 \):
\[ 0 = -\frac{1}{24}x^2 + 2x \] Factoring out \( x \): \[ 0 = x(-\frac{1}{24}x + 2) \]
This gives us one intercept at \( x = 0 \) (where the ball was hit) and the other we can solve for:
\[ -\frac{1}{24}x + 2 = 0 \] Solving for \( x \): \[ -\frac{1}{24}x = -2 \quad \Rightarrow \quad x = 48 \]
Thus, the x-intercepts occur at \( x = 0 \) (where the ball was hit) and \( x = 48 \) (where it lands).
Summary
- Vertex: The maximum height of the ball is 24 units at a horizontal distance of 24 units from home plate. This indicates the peak of the ball's trajectory.
- Intercepts: The ball starts its path at \( x = 0 \) when it was hit and lands at \( x = 48 \) units, showing how far the ball traveled before touching down.
These features help us understand the trajectory of the ball, including its highest point and the distance it traveled.