The standard form of a quadratic function is given by:

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

where $(h, k)$ is the vertex of the parabola. To convert $f(x) = x^2 + 8x$ into this form, we complete the square:

1. Start with the given function: $f(x) = x^2 + 8x$

2. Add and subtract $\left( \frac{8}{2} \right)^2 = 16$ inside the function: $f(x) = x^2 + 8x + 16 - 16$

3. Rewrite it as a perfect square: $f(x) = (x + 4)^2 - 16$

So, the standard form is:

$f(x) = (x + 4)^2 - 16$

This form shows that the vertex of the parabola is $(-4, -16)$.

Do you want more details or have any questions? Here are 5 related questions:

1. How do you find the vertex of a quadratic function?
2. What is the axis of symmetry for the function $f(x) = (x + 4)^2 - 16$?
3. How do you convert a quadratic function from standard form to general form?
4. Can you explain how to graph $f(x) = (x + 4)^2 - 16$?
5. What are the x-intercepts of the function $f(x) = x^2 + 8x$?

Tip: Completing the square is a helpful technique for converting a quadratic function into standard form, which makes it easier to identify the vertex and graph the function.