To analyze the function \( y = 2(x - 1)(x + 2) \), we can expand and simplify it:

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

2. **Identify key features**:
   - **Vertex**: The vertex form of a quadratic \( ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \).
     - Here, \( a = 2 \) and \( b = 2 \):
     \[
     x = -\frac{2}{2 \times 2} = -\frac{1}{2}
     \]
   - **Y-intercept**: Set \( x = 0 \):
     \[
     y = 2(0 - 1)(0 + 2) = 2(-1)(2) = -4
     \]
   - **X-intercepts**: Set \( y = 0 \):
     \[
     0 = 2(x - 1)(x + 2) \implies x - 1 = 0 \text{ or } x + 2 = 0 \implies x = 1 \text{ or } x = -2
     \]

3. **Summary of key features**:
   - Vertex: \( (-\frac{1}{2}, y) \) (substitute to find \( y \))
   - Y-intercept: \( (0, -4) \)
   - X-intercepts: \( (1, 0) \) and \( (-2, 0) \)

Would you like more details on any specific aspect or have any questions? Here are some follow-up questions:

1. What is the graph of the function \( y = 2(x - 1)(x + 2) \) like?
2. How can you find the vertex of any quadratic function?
3. What role do the intercepts play in graphing functions?
4. Can you provide the steps to factor similar quadratic equations?
5. How do transformations affect the graph of a quadratic function?

**Tip**: To find the vertex easily, you can use the formula \( x = -\frac{b}{2a} \) for any quadratic in standard form \( ax^2 + bx + c \).

Explore how to analyze and graph the quadratic function y = 2(x - 1)(x + 2). This problem includes expanding and simplifying the expression, finding the vertex, identifying intercepts, and understanding key graph features. Suitable for Grades 9-11 students studying quadratic functions.

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