The given quadratic function \( f(x) = - (x + 1.4)^2 + 6 \) describes the trajectory of a basketball shot. Let's analyze this function:
Vertex Form
The function is in vertex form, which is generally written as: \[ f(x) = a(x - h)^2 + k \] where \( (h, k) \) is the vertex of the parabola. In your function, we can see:
- \( a = -1 \) (indicating that the parabola opens downward).
- \( h = -1.4 \)
- \( k = 6 \)
Thus, the vertex of the parabola is: \[ (-1.4, 6) \] This means the highest point of the trajectory (the apex of the shot) occurs at a horizontal distance of -1.4 feet from the player (which could imply that this is the point at which the ball is at its highest after being shot), and at this distance, the height of the ball is 6 feet.
Maximum Height
Since \( a \) is negative, the maximum height of the ball is at the vertex \( (h, k) \). Therefore, the maximum height the ball reaches is 6 feet.
Horizontal Distance
The vertex suggests that the player is positioned at \( x = 0 \), and the peak of the ball's trajectory is reached at \( x = -1.4 \) feet. The function will produce real height values only for certain \( x \) values.
Finding the Roots
To determine how far the basketball travels horizontally when it hits the ground (height = 0), we can set the function equal to zero: \[ -(x + 1.4)^2 + 6 = 0 \] Solving for \( x \), we get: \[ -(x + 1.4)^2 = -6 \] \[ (x + 1.4)^2 = 6 \] Taking the square root of both sides, we have: \[ x + 1.4 = \pm \sqrt{6} \] So: \[ x = -1.4 \pm \sqrt{6} \]
Calculating the two values of \( x \):
- \( x = -1.4 + \sqrt{6} \)
- \( x = -1.4 - \sqrt{6} \)
Calculating the approximate numeric values, we note that \( \sqrt{6} \approx 2.45 \):
- \( x \approx -1.4 + 2.45 \approx 1.05 \)
- \( x \approx -1.4 - 2.45 \approx -3.85 \) (This value does not make sense in the context as it would be further away in the opposite direction from the player.)
Conclusion
In conclusion, the model predicts that:
- The maximum height of the ball is 6 feet, occurring at \( -1.4 \) feet horizontally from the shooter.
- The ball's trajectory will hit the ground at approximately \( 1.05 \) feet horizontally from the player.