A quadratic function has its vertex at the point \( (-5,-4) \). The function passes through the point \( (8,-3) \). When written in vertex form, the function is \( f(x)=a(x-h)^{2}+k \), where: \( a= \) \( h= \) \( k= \)

To find the quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), we start with the known vertex and a point that the function passes through. ### Step 1: Identify known values From the problem, we know: - The vertex \( (h, k) = (-5, -4) \) - The point \( (x, y) = (8, -3) \) ### Step 2: Substitute known values into the vertex form We can substitute \( h \) and \( k \) into the vertex form: \[ f(x) = a(x + 5)^2 - 4 \] ### Step 3: Use the point to find \( a \) Now we will use the point \( (8, -3) \) to find the value of \( a \). We substitute \( x = 8 \) and \( f(x) = -3 \) into the equation: \[ -3 = a(8 + 5)^2 - 4 \] ### Step 4: Simplify the equation Calculating \( (8 + 5)^2 \): \[ (8 + 5)^2 = 13^2 = 169 \] Now substitute this back into the equation: \[ -3 = a(169) - 4 \] ### Step 5: Solve for \( a \) Add 4 to both sides: \[ -3 + 4 = 169a \] \[ 1 = 169a \] Now divide by 169: \[ a = \frac{1}{169} \] ### Final Values Now we can summarize the values: - \( a = \frac{1}{169} \) - \( h = -5 \) - \( k = -4 \) Thus, the values are: - \( a = \frac{1}{169} \) - \( h = -5 \) - \( k = -4 \)