To model the path of a boomerang using the provided data, we can derive a quadratic function. A typical format for such a model could be a vertex form of a parabola, expressed as:
\[ y = a(x - h)^2 + k \]
Where \( (h, k) \) represents the vertex of the parabola. Based on the data points you provided (though it's incomplete), we need to determine the vertex and the constant \( a \).
Given Points:
- (0, 5.41)
- (4, 2.61)
- (10, 0.03)
From this data, we can choose to calculate the vertex based on the maximum height of the boomerang trajectory.
Finding the Vertex:
Let’s assume the vertex occurs between the values where the vertical distance is maximized (i.e., between the points). According to the vertical distances:
- At x = 0, y = 5.41
- At x = 4, y = 2.61
- At x = 10, y = 0.03
The value y = 5.41 appears to be the maximum height, so we can reasonably consider the vertex to be close to (0, 5.41).
Quadratic Equation:
To find a, we can use two points to calculate it. Let's use the point (4, 2.61):
Substituting \( (x, y) \) with (4, 2.61) gives:
\[ 2.61 = a(4 - 0)^2 + 5.41 \]
Calculating this, we have:
\[ 2.61 = 16a + 5.41 \] \[ 16a = 2.61 - 5.41 \] \[ 16a = -2.8 \] \[ a = -0.175 \]
Vertex Form:
Therefore, the equation in vertex form can be represented as:
\[ y = -0.175(x - 0)^2 + 5.41 \]
Simplifying gives:
\[ y = -0.175x^2 + 5.41 \]
Possible Response Choice:
From the choices you listed, we can see that:
- (x−24.73)=−3.83(y−2.76)²
- y=−3.83x²+21.14x−4.84
- (x−24.35)=−3.46(y−2.67)²
- y=−3.83x²+21.14x−4.84
- (x−2.49)=0.00(y−16.56)²
The second equation, which appears as \( y = -3.83x^2 + 21.14x - 4.84 \), could potentially fit a model; however, it does not match our derived values.
Since deriving equations directly from the numerical data points may vary, the assumption based on vertex and a calculated approach may lead us somewhere between the proposed options in your question. A fitting quadratic equation to the values derived may suffice; evaluating the options with your data context and intended endpoints would be better aligned.
If you have specific points for analysis or additional details, confirming based on polynomial fitting using technology can yield the expected outcomes effectively.