equation in vertex form for the parabola with focus \( \left(\frac{13}{2}, \frac{13}{4}\right) \) and directrix \( y=\frac{35}{4} \)

To find the equation of the parabola in vertex form, we start with the given focus \( \left(\frac{13}{2}, \frac{13}{4}\right) \) and directrix \( y = \frac{35}{4} \). The vertex is halfway between the focus and the directrix, yielding the vertex at \( \left(\frac{13}{2}, \frac{13}{4} + \frac{35}{4} / 2\right) = \left(\frac{13}{2}, \frac{24}{4}\right) = \left(\frac{13}{2}, 6\right) \). The parabola opens downward since the focus is below the directrix. The vertex form is given by \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex and \( p \) is the distance from the vertex to the focus. Here, \( p = 6 - \frac{13}{4} = \frac{24}{4} - \frac{13}{4} = \frac{11}{4} \). Thus, the equation is \( \left(x - \frac{13}{2}\right)^2 = -\frac{11}{4}(y - 6) \). Now, if you’re ever in doubt about the focus and directrix concepts in parabolas, think of them as the superhero team: the focus is like the hero (the point you want to be closest to), and the directrix is the villain (the line that nobody wants to be near). This teaming helps keep any point on the parabola equally distanced from both! If you're curious for more, grab a book on conic sections or dive into some online courses. Resources like "Geometry: A Comprehensive Course" or even Khan Academy offer great insights into the world of parabolas, helping you understand not only their shapes but also their real-world applications in engineering and design.