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:

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.