To find the vertex of the quadratic function $f(x) = x^2 - 2x - 24$, we can use the vertex formula for a parabola:

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

In the given equation $f(x) = x^2 - 2x - 24$, the coefficients are:

• $a = 1$
• $b = -2$

Now, substitute these values into the vertex formula:

$x = \frac{-(-2)}{2(1)} = \frac{2}{2} = 1$

So, the x-coordinate of the vertex is $x = 1$.

Now, substitute $x = 1$ back into the original function to find the y-coordinate of the vertex:

$f(1) = (1)^2 - 2(1) - 24 = 1 - 2 - 24 = -25$

Thus, the vertex is $(1, -25)$.

The correct answer is: (1, -25).

Would you like a more detailed explanation of the steps or have any other questions on this? Here are some follow-up questions:

1. How does the vertex formula work in general for parabolas?
2. What is the significance of the vertex in terms of the graph's symmetry?
3. Can we find the vertex for a quadratic equation in standard form using completing the square?
4. How does the sign of the coefficient $a$ affect the direction of the parabola?
5. Can you find the vertex for the equation $y = -3x^2 + 6x + 2$?

Tip: Always remember that the vertex formula gives you the x-coordinate, and then you can substitute it back into the equation to find the y-coordinate!