To rewrite $f(x) = x^2 - 8x + 15$ in vertex form, we complete the square. The vertex form of a quadratic function is:

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

Here are the steps:

1. Start with the given equation: $f(x) = x^2 - 8x + 15$

2. Group the quadratic and linear terms, and set up for completing the square: $f(x) = (x^2 - 8x) + 15$

3. Find the value to complete the square: Take half of the coefficient of $x$ (which is $-8$), square it, and add and subtract that value inside the parenthesis. Half of $-8$ is $-4$, and $(-4)^2 = 16$.

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

4. Simplify the equation: Combine the perfect square trinomial and the constants.

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

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

5. Final vertex form: $f(x) = (x - 4)^2 - 1$

Vertex:

The vertex of the quadratic function is $(h, k) = (4, -1)$.

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Related Questions:

1. How do you derive the vertex form of a quadratic function in general?
2. What is the significance of the vertex in a quadratic function?
3. How do you find the axis of symmetry for a quadratic equation?
4. How can the vertex form help in identifying the maximum or minimum value?
5. What is the process for converting vertex form back to standard form?

Tip:

Completing the square is a powerful tool not just for rewriting equations but also for solving quadratics and analyzing their properties.