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Let's solve the given quadratic equation step by step:

### Equation: 
\[ y = x^2 - 2x - 3 \]

#### Step 1: Identify the y-intercept
The y-intercept occurs when \( x = 0 \). Plug \( x = 0 \) into the equation:
\[
y = 0^2 - 2(0) - 3 = -3
\]
So, the y-intercept is \( (0, -3) \).

#### Step 2: Find the x-intercepts
To find the x-intercepts, set \( y = 0 \):
\[
0 = x^2 - 2x - 3
\]
Factor the quadratic equation:
\[
0 = (x - 3)(x + 1)
\]
This gives the solutions \( x = 3 \) and \( x = -1 \). Therefore, the x-intercepts are \( (3, 0) \) and \( (-1, 0) \).

#### Step 3: Find the vertex
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 1 \) and \( b = -2 \), so:
\[
x = -\frac{-2}{2 \cdot 1} = 1
\]
Now, plug \( x = 1 \) back into the original equation to find \( y \):
\[
y = 1^2 - 2(1) - 3 = 1 - 2 - 3 = -4
\]
So, the vertex is \( (1, -4) \).

### Summary:
- **Y-intercept**: \( (0, -3) \)
- **X-intercepts**: \( (3, 0) \) and \( (-1, 0) \)
- **Vertex**: \( (1, -4) \)

To graph the parabola, plot these points and use the shape of the parabola, which opens upwards (since the coefficient of \( x^2 \) is positive).

Would you like more details on graphing, or have any questions?

### Related Questions:
1. How do you find the axis of symmetry for a parabola?
2. What does the sign of the coefficient of \( x^2 \) indicate about the parabola's direction?
3. How would the graph change if the equation was \( y = -x^2 + 2x + 3 \)?
4. How can you use completing the square to find the vertex?
5. What are the properties of a parabola in vertex form, \( y = a(x - h)^2 + k \)?

### Tip:
When graphing parabolas, always start by identifying the vertex and axis of symmetry, as they help guide the shape of the curve.

Identify the y-intercept, the x intercepts, and the vertex of the quadratic equation y = x^2 - 2x - 3, and then graph the parabola.

Learn how to graph the quadratic equation y = x^2 - 2x - 3 by finding the y-intercept, x-intercepts, and the vertex. This algebra problem helps you master graphing parabolas and understanding quadratic functions, perfect for students in Grades 8-10.

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