Express y= -32x^2 -32x +3 in vertex form.

To express the quadratic function \( y = -32x^2 - 32x + 3 \) in vertex form, we use **completing the square**. The general vertex form of a quadratic function is:

\[
y = a(x - h)^2 + k
\]

Where \((h, k)\) is the vertex of the parabola. Let's complete the square for this equation step by step.

1. **Factor out the coefficient of \(x^2\):**

\[
y = -32(x^2 + x) + 3
\]

2. **Complete the square inside the parentheses:**

   - Take half of the coefficient of \(x\), which is \(1/2\), and square it: \((1/2)^2 = 1/4\).
   - Add and subtract \(1/4\) inside the parentheses:

\[
y = -32\left(x^2 + x + \frac{1}{4} - \frac{1}{4}\right) + 3
\]

\[
y = -32\left(\left(x + \frac{1}{2}\right)^2 - \frac{1}{4}\right) + 3
\]

3. **Simplify the expression:**

   - Distribute the \(-32\) across the terms:

\[
y = -32\left(x + \frac{1}{2}\right)^2 + 32 \cdot \frac{1}{4} + 3
\]

\[
y = -32\left(x + \frac{1}{2}\right)^2 + 8 + 3
\]

\[
y = -32\left(x + \frac{1}{2}\right)^2 + 11
\]

So, the vertex form of the quadratic equation is:

\[
y = -32\left(x + \frac{1}{2}\right)^2 + 11
\]

The vertex of the parabola is \(\left(-\frac{1}{2}, 11\right)\).

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Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do you find the vertex of a quadratic function directly from standard form?
2. What is the significance of the vertex in a quadratic function?
3. How does completing the square help in solving quadratic equations?
4. Can you convert a quadratic function from vertex form back to standard form?
5. How does the value of "a" in the vertex form affect the graph of a quadratic function?

**Tip:** When completing the square, always remember to balance the equation by adding and subtracting the same value!

Learn how to convert the quadratic function y = -32x^2 - 32x + 3 to vertex form using the method of completing the square. This step-by-step guide covers key concepts like quadratic functions and completing the square, suitable for students in Grades 9-11.

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