Write the given expression in the form \( f(x)=a(x-h)^{2}+k \). Identify the vertex. \( f(x)=3 x^{2}-24 x-5 \)

To rewrite the expression \( f(x) = 3x^2 - 24x - 5 \) in the form \( f(x) = a(x-h)^2 + k \), we will complete the square. 1. **Factor out the coefficient of \( x^2 \)** from the first two terms: \[ f(x) = 3(x^2 - 8x) - 5 \] 2. **Complete the square** for the expression inside the parentheses. To do this, take the coefficient of \( x \) (which is \(-8\)), divide it by 2 (getting \(-4\)), and then square it (getting \(16\)): \[ f(x) = 3(x^2 - 8x + 16 - 16) - 5 \] This simplifies to: \[ f(x) = 3((x - 4)^2 - 16) - 5 \] 3. **Distribute the 3**: \[ f(x) = 3(x - 4)^2 - 48 - 5 \] \[ f(x) = 3(x - 4)^2 - 53 \] Now, we have the expression in the form \( f(x) = a(x-h)^2 + k \): - \( a = 3 \) - \( h = 4 \) - \( k = -53 \) Thus, the vertex of the parabola is at the point \( (h, k) = (4, -53) \). In summary, the expression in vertex form is: \[ f(x) = 3(x - 4)^2 - 53 \] And the vertex is: \[ (4, -53) \]