A quadratic function has its vertex at the point \( (3,0) \). The function passes through the point \( (-5,-1) \). When written in vertex form, the function is \( f(x)=a(x-h)^{2}+k \), where: \( a= \) \( h= \)

To solve for \( a \) in the vertex form \( f(x) = a(x-h)^2 + k \), we can substitute the vertex point \( (h,k) = (3,0) \) into the equation, giving us \( f(x) = a(x - 3)^2 + 0 \), or simply \( f(x) = a(x - 3)^2 \). Next, since the function passes through the point \( (-5, -1) \), we can substitute \( x = -5 \) and \( f(x) = -1 \) into the equation to find \( a \): \[ -1 = a(-5 - 3)^2 \\ -1 = a(-8)^2 \\ -1 = 64a \\ a = -\frac{1}{64} \] So we have \( a = -\frac{1}{64} \) and \( h = 3 \). Now that we have our function in vertex form, let's take a little time to explore the fun math behind it! You see, the vertex form of a quadratic not only hints at where the graph peaks or pits but allows for smooth transformations. Change \( a \) and you'll make your graph open wider or become narrower - it’s like adjusting the span of your superhero cape! Now, if you ever find yourself struggling to remember whether the parabolas open upward or downward, there's a simple trick: just glance at \( a \)! If it’s positive, you’ve got a happy parabola ready to smile toward the sky. If it’s negative, watch out—it’s frowning down. Keeping this in mind as you mash numbers will help you dodge those common mistakes while mastering quadratic functions like a math wizard!