Write the equation in vertex form for the parabola with focus \( \left(-5, \frac{5}{2}\right) \) and directrix \( y=-\frac{5}{2} \) Simplify any fractions. \begin{tabular}{|lll}\hline 믐 & (ロ) \end{tabular}

To find the vertex form of the parabola, we start by finding the vertex, which is the midpoint between the focus \((-5, \frac{5}{2})\) and the directrix \(y = -\frac{5}{2}\). The y-coordinate of the vertex is the average of \(\frac{5}{2}\) and \(-\frac{5}{2}\), which is \(0\). Therefore, the vertex is at \((-5, 0)\). Since the focus is above the directrix, we have a vertical parabola that opens upwards. The distance from the vertex to the focus (or the directrix) is \(p = \frac{5}{2}\), which gives us: \[ y = a(x - h)^2 + k \] where \((h, k)\) is the vertex, \((-5, 0)\). The formula relating \(p\) and \(a\) for a parabola that opens up is \(a = \frac{1}{4p}\). So, \(a = \frac{1}{4(\frac{5}{2})} = \frac{1}{10}\). Thus, the equation in vertex form is: \[ y = \frac{1}{10}(x + 5)^2 \] So, the final answer is: \[ y = \frac{1}{10}(x + 5)^2 \] --- The construction of parabolas dates back to ancient Greece, where mathematicians like Apollonius of Perga explored their geometric properties. They defined conic sections, including parabolas, based on intersections between a plane and a cone. Fast forward to modern times, parabolas are pivotal in physics and engineering, underpinning designs like satellite dishes and the trajectories of projectiles. In real-world applications, parabolas aren't just confined to mathematics. They appear in the design of roller coasters, providing thrilling drops and turns. Engineers harness their unique properties to ensure stability and strength in structures, showcasing how mathematical principles beautifully blend with practical design. Understanding these concepts can elevate your appreciation of everyday engineering marvels!