The problem states the quadratic equation $y = x^2 - 2x - 1$, and asks to write it in vertex form.

Vertex Form of a Quadratic Equation:

The vertex form is written as: $y = a(x-h)^2 + k$ where $(h, k)$ is the vertex of the parabola.

To convert $y = x^2 - 2x - 1$ into vertex form, we use completing the square:

Step 1: Group the quadratic and linear terms

$y = (x^2 - 2x) - 1$

Step 2: Complete the square

Take half the coefficient of $x$, square it, and add/subtract it inside the parentheses.

The coefficient of $x$ is $-2$. Half of $-2$ is $-1$, and squaring it gives: $(-1)^2 = 1$

Now add and subtract $1$ inside the parentheses: $y = (x^2 - 2x + 1 - 1) - 1$

$y = ((x - 1)^2 - 1) - 1$

Step 3: Simplify

$y = (x - 1)^2 - 2$

Final Answer:

The equation in vertex form is: $y = (x - 1)^2 - 2$

The vertex is at $(1, -2)$.

Would you like further clarifications or additional examples?

Related Questions:

1. How do you find the vertex of a parabola given in standard form?
2. What is the axis of symmetry of the parabola $y = (x - 1)^2 - 2$?
3. How does completing the square relate to factoring quadratics?
4. What are some practical applications of the vertex form of a quadratic equation?
5. How do you graph a parabola given in vertex form?

Tip:

When completing the square, always check your work by expanding the vertex form back into standard form to ensure correctness!