To convert the quadratic equation \( y = -\frac{1}{3}x^2 + \frac{2}{3}x + 6 \) into vertex form, we need to complete the square. The vertex form of a quadratic equation is:

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

where \((h, k)\) is the vertex of the parabola.

### Step-by-Step Process:

1. **Factor out the coefficient of \(x^2\)** from the first two terms:
   \[
   y = -\frac{1}{3}\left(x^2 - 2x\right) + 6
   \]

2. **Complete the square**:
   To complete the square for the expression inside the parentheses, we take half of the coefficient of \(x\) (which is \(-2\)) and square it:
   \[
   \left(\frac{-2}{2}\right)^2 = 1
   \]
   Now, add and subtract 1 inside the parentheses:
   \[
   y = -\frac{1}{3}\left(x^2 - 2x + 1 - 1\right) + 6
   \]
   This simplifies to:
   \[
   y = -\frac{1}{3}\left((x - 1)^2 - 1\right) + 6
   \]

3. **Simplify the equation**:
   Distribute the \(-\frac{1}{3}\):
   \[
   y = -\frac{1}{3}(x - 1)^2 + \frac{1}{3} + 6
   \]
   Combine the constants:
   \[
   y = -\frac{1}{3}(x - 1)^2 + \frac{19}{3}
   \]

### Final vertex form:
\[
y = -\frac{1}{3}(x - 1)^2 + \frac{19}{3}
\]

So, the vertex of the parabola is \((1, \frac{19}{3})\).

Would you like more details or have any questions?

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Here are 5 related questions to explore further:

1. How do you find the vertex of a quadratic equation in standard form directly?
2. What is the significance of the vertex in a quadratic function?
3. Can you graph the function \( y = -\frac{1}{3}(x - 1)^2 + \frac{19}{3} \) to verify the vertex?
4. How does completing the square help in solving quadratic equations?
5. What is the axis of symmetry for the quadratic function you just converted?

**Tip:** Completing the square is a useful technique not only for rewriting quadratics but also for solving them!

turn y = -1/3x^2 + 2/3x + 6 to vertex form

Learn how to convert the quadratic equation y = -1/3x^2 + 2/3x + 6 into vertex form using the completing the square method. This step-by-step guide explains key algebraic concepts and is suitable for students in Grades 9-10.

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