equation in vertex form for the parabola with focus \( (0,-6) \) and directrix \( y=-10 \). any fractions.

To find the equation in vertex form, we start by determining the vertex's coordinates, which can be found as the midpoint between the focus and the directrix. The focus is at \( (0, -6) \) and the directrix is the line \( y = -10 \). The vertex will be at \( (0, -8) \), halfway between -6 and -10. The parabola opens upwards since the focus is above the directrix. Using the vertex form of a parabola, \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex, we substitute \( h = 0 \) and \( k = -8 \). The value of \( a \) can be calculated using the distance formula: \( p = \frac{1}{4a} \). Here, \( p \) is the distance from the vertex to the focus, which is 2 units, so \( a = \frac{1}{8} \). Thus, the equation is: \[ y = \frac{1}{8}x^2 - 8 \] --- The midpoint serves as an essential link in understanding the balance and shape of the parabola, leading to its vertex form. This kind of geometric insight is not just helpful in solving quadratic equations but can also enhance spatial reasoning in various math problems. The directrix and focus play a crucial role in defining parabolas and can be related to real-world applications, such as satellite dishes or car headlights, where the focus helps to direct light or signals. Understanding this can lead to innovative designs that leverage the properties of parabolas to optimize functionality in engineering and technology.