The function provided in the image is:

$f(x) = -x^2 + 10x - 23$

Step-by-Step Conversion to Vertex Form

The vertex form of a quadratic function is: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Step 1: Factor out $-1$ from the first two terms

The leading coefficient of the $x^2$-term is negative, so we factor $-1$ from the quadratic and linear terms:

$f(x) = -\left(x^2 - 10x\right) - 23$

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$ in the parentheses (which is $-10$), square it, and add/subtract it inside the parentheses.

$\text{Half of } -10 = -5, \quad (-5)^2 = 25$

Add and subtract $25$ inside the parentheses:

$f(x) = -\left(x^2 - 10x + 25 - 25\right) - 23$

$f(x) = -\left((x - 5)^2 - 25\right) - 23$

Step 3: Simplify the expression

Distribute the $-1$ and simplify:

$f(x) = -(x - 5)^2 + 25 - 23$

$f(x) = -(x - 5)^2 + 2$

Final Vertex Form:

$f(x) = -(x - 5)^2 + 2$

The vertex of the function is $(5, 2)$.

Would you like me to go over additional steps or provide a graph for better understanding?

Here are 5 related questions for practice:

1. How do you determine the vertex from the vertex form of a quadratic equation?
2. What are the steps to find the axis of symmetry for a parabola?
3. How does the sign of the coefficient $a$ affect the graph of a parabola?
4. Can you convert $f(x) = x^2 - 6x + 8$ into vertex form?
5. What is the significance of completing the square in algebra?

Tip: Always check your work by expanding the vertex form back into standard form!