Write the equation in vertex form for the parabola with focus \( (0,6) \) and directrix \( y=-6 \). Simplify any fractions. \begin{tabular}{|lll}\hline\( \frac{\square}{\square} \) & (ם) \end{tabular}

To rewrite the equation of the parabola in vertex form, we first identify the vertex, which is the midpoint between the focus \( (0, 6) \) and the directrix \( y = -6 \). The vertex is at \[ \left(0, \frac{6 + (-6)}{2}\right) = (0, 0). \] The distance from the vertex to the focus (or directrix) is \( p = 6 \), since the vertex is at the origin and the focus is at \( (0, 6) \). For a parabola that opens upwards, the vertex form is given by \[ y = a(x - h)^2 + k, \] where \( (h, k) \) is the vertex and \( a = \frac{1}{4p} \). In our case, \( h = 0 \) and \( k = 0 \), hence: \[ a = \frac{1}{4 \cdot 6} = \frac{1}{24}. \] Thus, the equation in vertex form is \[ y = \frac{1}{24} x^2. \] And there you have it! The beautiful parabola with a focus shining bright at \( (0, 6) \) and a sturdy directrix sitting comfortably at \( y = -6 \). Now, you're all clear on vertex forms, so can you imagine making a giant smiley face using parabolas in real life? Hang a large round paper below your vertex, and suddenly you've got the perfect cheeky grin! Just think about all the parabolas in architecture as they frame windows or arches, adding character and depth to structures all around us.