To analyze the equation \( y = 2(x - 1)(x + 2) \), we can expand it to better understand its characteristics.

1. **Expanding the equation**:
   \[
   y = 2(x - 1)(x + 2)
   \]
   \[
   = 2[(x)(x) + (2x) - (1x) - (2)]
   \]
   \[
   = 2(x^2 + x - 2)
   \]
   \[
   = 2x^2 + 2x - 4
   \]

2. **Characteristics**:
   - **Type**: This is a quadratic equation (a parabola).
   - **Vertex**: The vertex can be found using the formula \( x = -\frac{b}{2a} \):
     \[
     x = -\frac{2}{2 \cdot 2} = -\frac{1}{2}
     \]
   - **Y-intercept**: Set \( x = 0 \):
     \[
     y = 2(0 - 1)(0 + 2) = 2(-1)(2) = -4
     \]
   - **X-intercepts**: Set \( y = 0 \):
     \[
     0 = 2(x - 1)(x + 2)
     \]
     The x-intercepts are \( x = 1 \) and \( x = -2 \).

3. **Graph**: The parabola opens upwards (since the coefficient of \( x^2 \) is positive).

Would you like to explore any specific aspects of this equation further? Here are some related questions you might consider:

1. What is the vertex of the parabola?
2. How do you determine the direction the parabola opens?
3. What is the significance of the coefficients in the quadratic equation?
4. How can you graph this equation accurately?
5. What would the effects be if the coefficient of \( x^2 \) were negative?

**Tip**: When dealing with quadratic equations, always look for the vertex and intercepts to sketch the graph accurately!

Explore the characteristics of the quadratic equation y = 2(x - 1)(x + 2) with a step-by-step breakdown, including expansion, finding the vertex, intercepts, and graphing the parabola. Ideal for students in Grades 8-10 looking to understand parabolas and key algebraic concepts.

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