https://file.aiquickdraw.com/mathbot/file/1726703518482pemnk4w3.jpg

Let's analyze the problem by focusing on both the given functions \( f(x) = -2(x - 3)^2 + 2 \) and the graph for \( g(x) \).

### 1. **Understanding \( f(x) \):**
   - The function \( f(x) = -2(x - 3)^2 + 2 \) is a parabola with the following properties:
     - **Vertex**: The vertex occurs at \( x = 3 \), with \( f(3) = 2 \), meaning the vertex is \( (3, 2) \).
     - **Shape**: Since the coefficient of \( (x-3)^2 \) is negative (-2), the parabola opens downwards.
     - **Increasing/Decreasing**: 
       - It is **increasing** on the interval \( (-\infty, 3) \).
       - It is **decreasing** on the interval \( (3, \infty) \).
     - **Positive**: The function is above the x-axis (positive) when \( f(x) > 0 \), which occurs between the roots of the equation. The roots are approximately at \( x = 1 \) and \( x = 5 \), so the function is positive on the interval \( (1, 5) \).
     - **Negative**: The function is below the x-axis (negative) for \( x \in (-\infty, 1) \cup (5, \infty) \).

### 2. **Understanding \( g(x) \) from the Graph**:
   - From the graph, \( g(x) \) is another parabola opening downwards, similar to \( f(x) \).
     - **Vertex**: The vertex seems to be at \( (0, 2) \).
     - **Increasing/Decreasing**:
       - It is **increasing** on the interval \( (-\infty, 0) \).
       - It is **decreasing** on the interval \( (0, \infty) \).
     - **Positive**: The graph of \( g(x) \) is positive (above the x-axis) for \( (-2, 2) \).
     - **Negative**: The graph of \( g(x) \) is negative (below the x-axis) for \( (-\infty, -2) \cup (2, \infty) \).

### 3. **Assigning Intervals to Each Category:**

|                   | \( f(x) \)        | \( g(x) \)       |
|-------------------|-------------------|------------------|
| **Increasing**     | \( (-\infty, 3) \) | \( (-\infty, 0) \) |
| **Decreasing**     | \( (3, \infty) \) | \( (0, \infty) \) |
| **Positive**       | \( (1, 5) \)      | \( (-2, 2) \)     |
| **Negative**       | \( (-\infty, 1) \cup (5, \infty) \) | \( (-\infty, -2) \cup (2, \infty) \) |

Feel free to ask for more details or explanations on any step!

### Five relative questions:
1. How do we find the vertex of a parabola in general?
2. What is the significance of the parabola opening upwards or downwards?
3. How do we determine whether a quadratic function is increasing or decreasing?
4. How do we solve for the roots of a quadratic function like \( f(x) \)?
5. How can we check if a function is positive or negative over an interval?

### Tip:
Remember, a negative leading coefficient in a quadratic function always makes the parabola open downwards, indicating a maximum point at the vertex.

Consider the functions f(x) = -2(x - 3)^2 + 2 and g(x), which is represented by the graph. Describe key features such as increasing, decreasing, positive, and negative intervals for both f(x) and g(x).

Analyze the quadratic functions f(x) = -2(x - 3)^2 + 2 and g(x) represented by a graph. Learn about increasing, decreasing, positive, and negative intervals for these functions, ideal for students in Grades 10-12 studying quadratic functions and parabolas.

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