https://file.aiquickdraw.com/mathbot/file/1726544028901qy4m2lzf.jpg

The image contains a set of quadratic functions, and the task is to rewrite each one in standard form and find the vertex.

To convert the quadratic functions into standard form \( f(x) = a(x-h)^2 + k \), we will complete the square and identify the vertex \( (h, k) \).

I'll start solving these one by one. Let's begin with the first function:

### 6. \( f(x) = x^2 - 12x + 32 \)

1. The equation is \( f(x) = x^2 - 12x + 32 \).
2. To complete the square, take half the coefficient of \(x\), square it, and add/subtract it inside the equation.
   - Half of \(-12\) is \(-6\), and \((-6)^2 = 36\).
   - So, rewrite the function as:
     \[
     f(x) = (x^2 - 12x + 36) - 36 + 32 = (x - 6)^2 - 4
     \]
3. The standard form is \( f(x) = (x - 6)^2 - 4 \).
4. The vertex is \( (h, k) = (6, -4) \).

I'll now continue with the rest.

### 7. \( g(x) = x^2 + 2x - 3 \)

1. The equation is \( g(x) = x^2 + 2x - 3 \).
2. Completing the square:
   - Half of 2 is 1, and \( (1)^2 = 1 \).
   - Rewrite the function:
     \[
     g(x) = (x^2 + 2x + 1) - 1 - 3 = (x + 1)^2 - 4
     \]
3. The standard form is \( g(x) = (x + 1)^2 - 4 \).
4. The vertex is \( (-1, -4) \).

### 8. \( f(x) = x^2 - x \)

1. The equation is \( f(x) = x^2 - x \).
2. Completing the square:
   - Half of \(-1\) is \(-\frac{1}{2}\), and \( \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \).
   - Rewrite the function:
     \[
     f(x) = \left(x^2 - x + \frac{1}{4}\right) - \frac{1}{4} = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4}
     \]
3. The standard form is \( f(x) = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} \).
4. The vertex is \( \left(\frac{1}{2}, -\frac{1}{4}\right) \).

Would you like me to continue with the remaining problems? Let me know if you'd like details on any step.

Rewrite the quadratic functions in standard form and give the vertex.

This page provides a step-by-step solution for rewriting quadratic functions in standard form and finding the vertex. Problems include equations like x^2 - 12x + 32 and x^2 + 2x - 3, using completing the square method to find the vertex form.

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