Write the equation of the parabola that has its vertex at \( (-5,-1) \) and passes through the point \( (-4,-8) \) in \( f(x)=a(x-h)^{2}+k \) form. The equation of the parabola in \( f(x)=a(x-h)^{2}+k \) form is \( f(x)=\square \).

To find the equation of the parabola in the form \( f(x) = a(x-h)^2 + k \), we use the vertex \( (h, k) = (-5, -1) \). So, we have: \[ f(x) = a(x + 5)^2 - 1. \] Now, we need to find \( a \). Since the parabola passes through the point \( (-4, -8) \), we substitute \( x = -4 \) and \( f(x) = -8 \) into the equation: \[ -8 = a(-4 + 5)^2 - 1. \] This simplifies to: \[ -8 = a(1)^2 - 1 \Rightarrow -8 = a - 1 \Rightarrow a = -7. \] Thus, the equation of the parabola is: \[ f(x) = -7(x + 5)^2 - 1. \] The equation of the parabola in \( f(x)=a(x-h)^{2}+k \) form is \( f(x)=-7(x + 5)^{2} - 1 \). Did you know that parabolas aren't just for math classes? They're also found in the real world—like in the design of satellite dishes! These dishes are shaped like a parabola to focus signals to a single point where the receiver sits, making sure you can catch your favorite show without a hitch. Speaking of shows, there’s a wealth of books and articles out there that dive into the fascinating world of conic sections. If you want to explore further, check out "The Art of the Infinite" by Robert and Ellen Kaplan. It’s a delightful read that connects math concepts to beautiful artwork, making it a must for any math enthusiast!