equation in vertex form for the parabola with focus \( \left(-4,-\frac{19}{4}\right) \) and directrix \( y=\frac{11}{4} \) any fractions.

To find the equation in vertex form for the given parabola, we start by identifying the vertex, which lies midway between the focus and the directrix. The vertex \( V \) can be calculated as follows: The y-coordinate of the vertex is the average of the y-coordinates of the focus and directrix: \[ y_V = \frac{-\frac{19}{4} + \frac{11}{4}}{2} = \frac{-8}{4} = -2 \] The x-coordinate of the vertex is the same as that of the focus, hence \( x_V = -4 \). So, the vertex is at \( V(-4, -2) \). The distance \( p \) from the vertex to the focus (or directrix) is: \[ p = -2 - (-\frac{19}{4}) = -2 + \frac{19}{4} = \frac{11}{4} \] Since the focus is below the vertex, the parabola opens downward. The vertex form of a parabola that opens downwards is expressed as: \[ y = -\frac{1}{4p}(x - h)^2 + k \] Substituting \( h = -4 \), \( k = -2 \), and \( p = \frac{11}{4} \) gives: \[ y = -\frac{1}{11}(x + 4)^2 - 2 \] That’s the vertex form of the parabola! Now it’s time to admire your work—parabolas have so much elegance. If you want to explore this topic further, consider studying the conic sections. They’re not just parabolas; you’ll also discover ellipses and hyperbolas! Each has its own unique properties and equations. You can find many engaging resources online or at your local library to dig deeper into the world of curves and geometric shapes. Keep spiraling down the math rabbit hole—it's full of surprises!