The given quadratic equation is:

$y = -2x^2 + 3x - 1$

We are asked to rewrite this equation in vertex form, which is:

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

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

Step 1: Completing the Square

First, we'll complete the square to rewrite the quadratic in vertex form.

1. Factor out the coefficient of $x^2$ from the first two terms:

$y = -2(x^2 - \frac{3}{2}x) - 1$

2. Complete the square:
   • Take half of the coefficient of $x$ inside the parentheses, $\frac{-3}{2}$, which is $\frac{-3}{4}$.
   • Square this value: $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$.

Now add and subtract this inside the parentheses to complete the square:

$y = -2\left(x^2 - \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) - 1$

3. Simplify:
   • Combine the perfect square and adjust the constant outside:

$y = -2\left[\left(x - \frac{3}{4}\right)^2 - \frac{9}{16}\right] - 1$

4. Distribute:
   • Distribute the $-2$ to both terms inside the brackets:

$y = -2\left(x - \frac{3}{4}\right)^2 + \frac{18}{16} - 1$

5. Simplify constants:
   • Simplify the constant terms:

$y = -2\left(x - \frac{3}{4}\right)^2 + \frac{18}{16} - \frac{16}{16}$

$y = -2\left(x - \frac{3}{4}\right)^2 + \frac{2}{16}$

$y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}$

Step 2: Identify the vertex

Now that we have the equation in vertex form:

$y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}$

The vertex $(h, k)$ is $\left(\frac{3}{4}, \frac{1}{8}\right)$.

Final Answer:

• Vertex form: $y = -2\left(x - \frac{3}{4}\right)^2 + \frac{1}{8}$
• Vertex: $\left(\frac{3}{4}, \frac{1}{8}\right)$

Would you like further explanation or details on any of the steps?

Here are some related questions you might consider:

1. What is the axis of symmetry for this quadratic equation?
2. How does the value of $a = -2$ affect the shape of the parabola?
3. How do you determine the direction in which the parabola opens?
4. Can you find the y-intercept of the original quadratic equation?
5. How do you find the x-intercepts of the quadratic equation?

Tip: Completing the square is a helpful technique not only for finding the vertex form of a quadratic equation but also for solving quadratic equations algebraically.