- Examine the symmetry of the given data points to identify that the axis of symmetry is \(x = 105\).
- Determine the vertex \((h, k) = (105, 7)\) by locating the minimum y-value in the table.
- Use the vertex form of the quadratic equation \(y = a(x - h)^2 + k\) and substitute a point to calculate the constant \(a \approx 0.0018\).
- Conclude that the vertex is \(\boxed{(105, 7)}\) and the model is \(\boxed{y = 0.0018(x - 105)^2 + 7}\).
Explanation
- Analyzing the Symmetry and Identifying the Vertex
The given data provides a set of coordinates that follow a parabolic trajectory. By examining the table of values:
- At \(x = 0\), \(y = 27\).
- At \(x = 52.5\), \(y = 12\).
- At \(x = 105\), \(y = 7\).
- At \(x = 157.5\), \(y = 12\) (assuming the value \(157.6\) is a slight measurement or typographical deviation from the symmetric point).
- At \(x = 210\), \(y = 27\).
Notice that the y-values are symmetric about the center value \(x = 105\). For example, at \(x = 0\) and \(x = 210\), the output is \(27\). Since the parabola is symmetric and reaches its minimum value of \(7\) at \(x = 105\), the vertex \((h, k)\) is \((105, 7)\).
- Deriving the Quadratic Equation
The vertex form of a quadratic equation is given by:
\[y = a(x - h)^2 + k\]
Substituting the vertex \((h, k) = (105, 7)\) into this form, we obtain:
\[y = a(x - 105)^2 + 7\]
To determine the value of the constant \(a\), we use one of the other data points, such as \((0, 27)\):
\[27 = a(0 - 105)^2 + 7\]
Subtracting \(7\) from both sides:
\[20 = a(-105)^2\]
\[20 = a(11025)\]
Solving for \(a\):
\[a = \frac{20}{11025} \approx 0.001814\]
Rounding to four decimal places gives \(a \approx 0.0018\). Thus, the equation modeling the situation is \(y = 0.0018(x - 105)^2 + 7\).
- Finalizing the Results
Based on the analysis, the vertex of the parabola is \((105, 7)\), where the x-coordinate \(h = 105\) and the y-coordinate \(k = 7\). The quadratic model that best fits the given points is \(y = 0.0018(x - 105)^2 + 7\). Therefore:
- The vertex is \(\boxed{(105, 7)}\).
- The correct model is \(\boxed{y = 0.0018(x - 105)^2 + 7}\).
Examples
In physics, parabolas often describe the trajectory of an object thrown into the air, such as a ball. By knowing the peak height (or the minimum point, in some mechanical designs like cable-stayed bridges) and a few other points along the path, engineers and scientists can determine the precise equation of motion. This allows them to predict where the object will land or determine the tension required in support cables to maintain the desired shape, ensuring structural integrity and safety.