To find the vertex of the quadratic function $y = 4x^2 - 24x + 11$, we can use the formula for the x-coordinate of the vertex in a quadratic equation of the form $y = ax^2 + bx + c$. The formula is:

$x = \frac{-b}{2a}$

Here, $a = 4$ and $b = -24$, so the x-coordinate of the vertex is:

$x = \frac{-(-24)}{2(4)} = \frac{24}{8} = 3$

Now, substitute $x = 3$ into the original equation to find the y-coordinate:

$y = 4(3)^2 - 24(3) + 11$ $y = 4(9) - 72 + 11$ $y = 36 - 72 + 11 = -25$

Therefore, the vertex of the function is $(3, -25)$.

Would you like any further explanation or details about how to work with quadratic functions?

Here are some follow-up questions:

1. How does the vertex formula relate to the graph of a parabola?
2. What does the coefficient $a$ in a quadratic function indicate about the direction of the parabola?
3. How can we determine if the vertex is a minimum or maximum point?
4. What is the importance of the discriminant in solving quadratic equations?
5. How do you graph a quadratic equation?

Tip: When trying to graph a quadratic equation, the vertex is key to drawing the curve correctly, as it represents the highest or lowest point on the parabola.