We are analyzing the quadratic function $y = 2(x+3)(x-1)$. Here's how to approach each question:

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Step 1: Determine if the parabola opens up or down

The parabola opens up if the coefficient of $x^2$ is positive, and opens down if it is negative.

1. Rewrite the equation into standard form by expanding: $y = 2(x+3)(x-1)$ Expand: $y = 2(x^2 + 2x - 3)$ Simplify: $y = 2x^2 + 4x - 6$ The coefficient of $x^2$ is $2$, which is positive. Therefore, the parabola opens up.

Step 2: Is the vertex a maximum or minimum?

• Since the parabola opens up, the vertex is the lowest point. The vertex represents a minimum value.

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Step 3: Find the vertex

To find the vertex, we use the formula for the $x$-coordinate of the vertex: $x = -\frac{b}{2a}$ From $y = 2x^2 + 4x - 6$:

• $a = 2$, $b = 4$, $c = -6$ $x = -\frac{4}{2(2)} = -\frac{4}{4} = -1$

To find the $y$-coordinate of the vertex, substitute $x = -1$ into the equation: $y = 2(-1)^2 + 4(-1) - 6$ $y = 2(1) - 4 - 6 = 2 - 4 - 6 = -8$ The vertex is $(-1, -8)$.

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Step 4: Find the x-intercepts

The x-intercepts occur when $y = 0$. Using the factored form $y = 2(x+3)(x-1)$, set $y = 0$: $2(x+3)(x-1) = 0$ $x+3 = 0 \quad \text{or} \quad x-1 = 0$ $x = -3 \quad \text{and} \quad x = 1$ The x-intercepts are $(-3, 0)$ and $(1, 0)$.

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Step 5: Find the axis of symmetry

The axis of symmetry is the vertical line passing through the $x$-coordinate of the vertex: $\text{Axis of symmetry: } x = -1$

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Step 6: Domain and range

1. Domain: Since this is a quadratic function, the domain is all real numbers: $\text{Domain: } (-\infty, \infty)$
2. Range: Since the parabola opens up and the vertex is the minimum point at $y = -8$, the range is: $\text{Range: } [-8, \infty)$

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Summary of Results:

1. Opens up (since $a > 0$).
2. Vertex is $(-1, -8)$.
3. Vertex is a minimum.
4. X-intercepts: $(-3, 0)$ and $(1, 0)$.
5. Axis of symmetry: $x = -1$.
6. Domain: $(-\infty, \infty)$.
7. Range: $[-8, \infty)$.

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Additional Notes:

To determine "up" or "down" and "min" or "max":

• Look at the coefficient of $x^2$: if positive, the parabola opens up (minimum). If negative, it opens down (maximum).

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Do you have any questions or need further details? Here are related questions:

1. How do we find the vertex if the equation is not in standard form?
2. Can we graph this function to visualize the results?
3. How is the range determined based on the vertex?
4. How do we find x-intercepts for functions not in factored form?
5. How does the axis of symmetry relate to the vertex?

Tip: To check your work, graph the function and confirm the vertex, intercepts, and symmetry visually.