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Let's analyze the quadratic equation given:

\[ y = x^2 - 2x - 8 \]

### 1. **Y-intercept:**
The y-intercept occurs when \( x = 0 \).
Substitute \( x = 0 \) into the equation:
\[
y = (0)^2 - 2(0) - 8 = -8
\]
Thus, the y-intercept is \( (0, -8) \).

### 2. **X-intercepts:**
The x-intercepts occur when \( y = 0 \). Set the equation to 0 and solve for \( x \):
\[
0 = x^2 - 2x - 8
\]
This is a quadratic equation. We can solve it using the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2}
\]
\[
x = \frac{2 \pm 6}{2}
\]
This gives two solutions:
\[
x = \frac{2 + 6}{2} = 4 \quad \text{and} \quad x = \frac{2 - 6}{2} = -2
\]
Thus, the x-intercepts are \( (4, 0) \) and \( (-2, 0) \).

### 3. **Vertex:**
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x_{\text{vertex}} = \frac{-b}{2a}
\]
For \( y = x^2 - 2x - 8 \), \( a = 1 \) and \( b = -2 \), so:
\[
x_{\text{vertex}} = \frac{-(-2)}{2(1)} = \frac{2}{2} = 1
\]
To find the corresponding \( y \)-coordinate, substitute \( x = 1 \) into the equation:
\[
y = (1)^2 - 2(1) - 8 = 1 - 2 - 8 = -9
\]
Thus, the vertex is \( (1, -9) \).

### 4. **Graphing:**
- The vertex is at \( (1, -9) \).
- The parabola passes through the x-intercepts \( (-2, 0) \) and \( (4, 0) \).
- The y-intercept is \( (0, -8) \).

Would you like a detailed graph or further explanations on any part of this problem?

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### Related Questions:
1. How do we calculate the axis of symmetry for a parabola?
2. What is the significance of the discriminant in the quadratic formula?
3. How do we determine if a parabola opens upwards or downwards?
4. How does completing the square help in identifying the vertex form of a quadratic?
5. How do you convert a quadratic equation from standard form to vertex form?

#### Tip:
Always check if the quadratic can be factored easily before using the quadratic formula to save time!

Identify the y-intercept, the x-intercepts, the vertex, and then graph: y = x^2 - 2x - 8.

Solve and graph the quadratic equation y = x^2 - 2x - 8 by finding the y-intercept, x-intercepts, and vertex. Learn key algebra concepts including the quadratic formula and graphing parabolas. Ideal for students in Grades 8-10.

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