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

To find the vertex form of the parabola, we first need to determine the vertex. The vertex is located halfway between the focus and the directrix. So, we calculate the vertex's y-coordinate by averaging the y-coordinates of the focus and the directrix: \[ \text{Vertex } y = \frac{10 + (-6)}{2} = \frac{4}{2} = 2 \] Thus, the vertex is \( (0, 2) \). Now, the distance \( p \) from the vertex to the focus (or to the directrix) is: \[ p = 10 - 2 = 8 \] Since the focus is above the vertex, the parabola opens upwards, giving us the vertex form: \[ y = a(x-h)^2 + k \] where \( (h, k) \) is the vertex \( (0, 2) \), and \( a = \frac{1}{4p} = \frac{1}{4 \times 8} = \frac{1}{32} \). Putting all this together, the equation of the parabola in vertex form is: \[ y = \frac{1}{32}(x - 0)^2 + 2 \] or simply: \[ y = \frac{1}{32}x^2 + 2 \] Now, let's dive into some fun additional information! Did you know that parabolas are not just confined to mathematics but pop up in real life too? From the path of thrown objects to satellite dishes that collect signals, their unique shape plays a crucial role in optics and engineering. Ever taken a ride on a roller coaster? The curves often resemble parabolic shapes for a smoother ride! When working with parabolas, a common mistake is neglecting to check whether you’ve placed the focus above or below the directrix. This changes the 'a' value's sign! Remember, if the focus is above, it's an upward-opening parabola; if below, it opens downward. This little detail can save you from some serious headaches in graphing!