To analyze the equation $y = 2(x - 1)(x + 2)$, we can expand and explore its properties:

Step 1: Expand the equation

[ y = 2[(x - 1)(x + 2)] = 2[x^2 + 2x - x - 2] = 2[x^2 + x - 2] = 2x^2 + 2x - 4 ]

Step 2: Identify the properties

1. Type: This is a quadratic function.

2. Standard Form: The standard form is $y = ax^2 + bx + c$ where $a = 2$, $b = 2$, and $c = -4$.

3. Vertex: The vertex can be found using $x = -\frac{b}{2a}$: $x = -\frac{2}{2 \times 2} = -\frac{1}{2}$ Substitute $x = -\frac{1}{2}$ back into the equation to find $y$: $y = 2\left(-\frac{1}{2} - 1\right)\left(-\frac{1}{2} + 2\right) = 2\left(-\frac{3}{2}\right)\left(\frac{3}{2}\right) = -\frac{9}{2}$ So, the vertex is at $\left(-\frac{1}{2}, -\frac{9}{2}\right)$.

4. Roots: Set $y = 0$: $0 = 2(x - 1)(x + 2) \implies x - 1 = 0 \quad \text{or} \quad x + 2 = 0$ Thus, the roots are $x = 1$ and $x = -2$.

5. Y-intercept: Set $x = 0$: $y = 2(0 - 1)(0 + 2) = 2(-1)(2) = -4$

Summary

• Expanded Form: $y = 2x^2 + 2x - 4$
• Vertex: $\left(-\frac{1}{2}, -\frac{9}{2}\right)$
• Roots: $x = 1$ and $x = -2$
• Y-intercept: $(0, -4)$

Would you like more details or do you have any questions? Here are some related questions you might consider:

1. What are the implications of the vertex on the graph?
2. How does changing the coefficient of $x^2$ affect the graph?
3. Can you find the axis of symmetry for this parabola?
4. How would you graph this function?
5. What other forms can this quadratic function take (e.g., vertex form)?

Tip: Always plot the vertex and intercepts to get a clearer picture of the quadratic's shape.