We are given the quadratic function:

$f(x) = 3x^2 - 24x + 50$

To rewrite this function in vertex form, we need to complete the square. The vertex form of a quadratic function is:

$f(x) = a(x-h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

Steps to complete the square:

1. Factor out the coefficient of $x^2$ from the first two terms: $f(x) = 3(x^2 - 8x) + 50$

2. Complete the square:

   • Take half of the coefficient of $x$ (which is $-8$), square it, and add inside the parentheses: $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$
   • Add and subtract 16 inside the parentheses: $f(x) = 3(x^2 - 8x + 16 - 16) + 50$
   • Rewrite this as: $f(x) = 3((x - 4)^2 - 16) + 50$

3. Simplify:

   • Distribute the 3: $f(x) = 3(x - 4)^2 - 48 + 50$
   • Combine constants: $f(x) = 3(x - 4)^2 + 2$

Thus, the vertex form of the function is:

$f(x) = 3(x - 4)^2 + 2$

The vertex of the parabola is $(4, 2)$.

Would you like more details on the process or have any questions?

Here are some related questions to deepen your understanding:

1. How can you find the axis of symmetry from the vertex form?
2. What is the significance of the value of "a" in the vertex form of a quadratic function?
3. How would you find the y-intercept of this quadratic function from the vertex form?
4. How can you convert the function from vertex form back to standard form?
5. How does the vertex form help in graphing the function?

Tip: Completing the square is a method that not only helps convert quadratics to vertex form but also provides a useful way to solve quadratic equations!