To convert the quadratic expression $-x^2 + 3x - 2$ into vertex form, we follow the process of completing the square.

1. Start with the given equation: $y = -x^2 + 3x - 2$

2. Factor out $-1$ from the $x$-terms: $y = -(x^2 - 3x) - 2$

3. Complete the square inside the parentheses:

   • Take half of the coefficient of $x$ (which is $-3$), square it: $\left(\frac{-3}{2}\right)^2 = \frac{9}{4}$.
   • Add and subtract $\frac{9}{4}$ inside the parentheses.

   $y = -(x^2 - 3x + \frac{9}{4} - \frac{9}{4}) - 2$

4. Simplify the expression:

   • The terms inside the parentheses now form a perfect square trinomial: $y = -\left( \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} \right) - 2$
   • Distribute the $-1$ to both terms inside the parentheses: $y = -\left(x - \frac{3}{2}\right)^2 + \frac{9}{4} - 2$
   • Simplify the constants: $y = -\left(x - \frac{3}{2}\right)^2 - \frac{8}{4}$ $y = -\left(x - \frac{3}{2}\right)^2 - 2$

So, the vertex form of $-x^2 + 3x - 2$ is: $y = -\left(x - \frac{3}{2}\right)^2 - 2$

The vertex of the parabola is $\left(\frac{3}{2}, -2\right)$.

Would you like more details, or do you have any questions?

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Here are 5 questions that expand on this concept:

1. How do you determine whether a parabola opens upwards or downwards?
2. What is the significance of the vertex in a quadratic equation?
3. Can the vertex form be converted back to standard form?
4. How does completing the square help in graphing parabolas?
5. How can the axis of symmetry be determined from the vertex form?

Tip: In vertex form, $y = a(x-h)^2 + k$, the vertex is always $(h, k)$, and $a$ determines the direction and width of the parabola.