To convert the quadratic function $f(x) = x^2 + 8x + 15$ into vertex form, we need to complete the square.

Here are the steps:

Step 1: Start with the given quadratic function

$f(x) = x^2 + 8x + 15$

Step 2: Focus on the $x^2 + 8x$ part

Leave the constant $+15$ for now, and just work on completing the square for $x^2 + 8x$.

$f(x) = (x^2 + 8x) + 15$

Step 3: Complete the square

To complete the square, take half of the coefficient of $x$ (which is 8), square it, and then add and subtract that value inside the function.

Half of 8 is 4, and $4^2 = 16$. So, add and subtract 16 inside the parentheses:

$f(x) = (x^2 + 8x + 16 - 16) + 15$

$f(x) = ((x + 4)^2 - 16) + 15$

Step 4: Simplify the expression

Now simplify by combining the constants $-16 + 15$.

$f(x) = (x + 4)^2 - 1$

Final Answer:

The vertex form of the function is: $f(x) = (x + 4)^2 - 1$

In this form, the vertex is $(-4, -1)$.

Do you have any questions or would you like further details?

Here are 5 related questions:

1. How do you find the vertex from the standard form of a quadratic?
2. How do you convert from vertex form back to standard form?
3. Can you complete the square if the coefficient of $x^2$ is not 1?
4. How do you find the axis of symmetry for a quadratic function?
5. What is the relationship between the vertex form and transformations of a parabola?

Tip: Always check if the quadratic term has a coefficient of 1 before completing the square. If not, factor it out first!