To convert the quadratic equation $y = x^2 - 2x - 3$ into the form $y = (x - h)^2 + k$, we need to complete the square. Here's how to do it step-by-step:

Step 1: Rewrite the equation

We begin with the equation: $y = x^2 - 2x - 3$ Focus on the quadratic and linear terms: $x^2 - 2x$.

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$ (which is -2), square it, and add and subtract that value inside the equation.

$\text{Half of } -2 \text{ is } -1, \quad (-1)^2 = 1$

So, we add and subtract 1: $y = (x^2 - 2x + 1) - 1 - 3$ $y = (x - 1)^2 - 4$

Step 3: Write in vertex form

Now the equation is in the form: $y = (x - 1)^2 - 4$

Thus, the answer is: $y = (x - 1)^2 - 4$

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. How do you determine the vertex of a quadratic function in vertex form?
2. How does completing the square help in solving quadratic equations?
3. Can you graph the equation $y = (x - 1)^2 - 4$ and explain the transformations?
4. How would you convert a quadratic equation from standard form to factored form?
5. What are the applications of vertex form in real-world problems?

Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value within the expression.