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The given equation is:

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

This represents a quadratic function of the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -2 \), and \( c = -2 \).

### Key features of the quadratic equation:

1. **Vertex**: The vertex of a parabola can be found using the formula \( x = -\frac{b}{2a} \). In this case:
   \[
   x = -\frac{-2}{2(1)} = 1
   \]
   To find the \( y \)-coordinate of the vertex, substitute \( x = 1 \) back into the equation:
   \[
   y = (1)^2 - 2(1) - 2 = 1 - 2 - 2 = -3
   \]
   So, the vertex is \( (1, -3) \).

2. **Axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex, given by \( x = 1 \).

3. **Y-intercept**: To find the y-intercept, set \( x = 0 \) in the equation:
   \[
   y = 0^2 - 2(0) - 2 = -2
   \]
   So, the y-intercept is \( (0, -2) \).

4. **X-intercepts**: To find the x-intercepts, solve the quadratic equation \( x^2 - 2x - 2 = 0 \). Using the quadratic formula:
   \[
   x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)}
   \]
   Simplifying:
   \[
   x = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}
   \]
   So, the x-intercepts are \( x = 1 + \sqrt{3} \) and \( x = 1 - \sqrt{3} \).

5. **Graph**: The graph is a parabola that opens upwards (since \( a = 1 > 0 \)).

Would you like more details on any part of this analysis?

### Related Questions:
1. How do we complete the square for this quadratic function?
2. How do we derive the formula for the vertex of a quadratic function?
3. What is the significance of the discriminant in determining the number of real roots?
4. How would the graph change if the coefficient of \( x^2 \) were negative?
5. How can we find the maximum or minimum value of a quadratic function?

**Tip**: The vertex form of a quadratic function can make it easier to graph and identify key features such as the vertex and the direction the parabola opens.

Learn how to solve the quadratic equation y = x^2 - 2x - 2 with step-by-step guidance. This problem involves key concepts such as finding the vertex, intercepts, and solving using the quadratic formula. Suitable for students in Grades 9-10.

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