https://file.aiquickdraw.com/mathbot/file/17295426687869dk96i2b.jpg

The equation provided in the image is:

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

This represents a quadratic equation, which describes a parabola. The general form of a quadratic equation is:

\[
y = ax^2 + bx + c
\]

In this case:
- \( a = 1 \)
- \( b = -2 \)
- \( c = -2 \)

### Key Features of this Quadratic Function:

1. **Vertex**: The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x_{\text{vertex}} = -\frac{b}{2a}
\]

Substitute \( a = 1 \) and \( b = -2 \):

\[
x_{\text{vertex}} = -\frac{-2}{2(1)} = 1
\]

To find the y-coordinate of the vertex, substitute \( x = 1 \) into the equation:

\[
y_{\text{vertex}} = (1)^2 - 2(1) - 2 = 1 - 2 - 2 = -3
\]

So, the vertex is \( (1, -3) \).

2. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex. The equation for the axis of symmetry is:

\[
x = 1
\]

3. **Direction of Opening**: Since the coefficient of \( x^2 \) is positive (\( a = 1 \)), the parabola opens upwards.

4. **Y-Intercept**: The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = (0)^2 - 2(0) - 2 = -2
\]

Thus, the y-intercept is \( (0, -2) \).

5. **X-Intercepts**: To find the x-intercepts, set \( y = 0 \) and solve for \( x \):

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

Solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute \( a = 1 \), \( b = -2 \), and \( c = -2 \):

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)} = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2}
\]

Simplifying:

\[
x = 1 \pm \sqrt{3}
\]

Thus, the x-intercepts are:

\[
x = 1 + \sqrt{3} \quad \text{and} \quad x = 1 - \sqrt{3}
\]

These are the solutions for the x-intercepts.

Would you like more details or have any specific questions about this problem?

### Here are 5 related questions to expand your understanding:
1. How would you rewrite the quadratic equation in vertex form?
2. Can you sketch the graph of the quadratic function based on the information given?
3. How can you verify the x-intercepts by graphing?
4. What role does the discriminant \( \Delta = b^2 - 4ac \) play in determining the number of x-intercepts?
5. How does changing the value of \( a \) affect the width of the parabola?

**Tip**: When solving quadratic equations, the discriminant is a quick way to check how many real solutions exist: if \( \Delta > 0 \), there are two distinct real solutions; if \( \Delta = 0 \), there's one real solution; and if \( \Delta < 0 \), no real solutions exist.

Solve the quadratic equation y = x^2 - 2x - 2 with a step-by-step solution. This covers algebraic concepts such as vertex, axis of symmetry, and x-intercepts. Suitable for high school students.

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