Let's analyze the graph in the image and answer the questions step by step.

1. Identifying the minimum of the function:

From the graph, we see that the curve reaches its lowest point at $(x, y) = (-4, -4)$, indicating that the function has a minimum value of -4 at $x = -4$.

2. Identifying where the function is increasing:

The function increases when moving from left to right, as the graph rises. Looking at the graph:

• The function increases from $x = -4$ to $x = 0$.

Thus, the function is increasing on the interval $(-4, 0)$.

3. Identifying where the function is decreasing:

The function decreases when the graph is falling. Looking at the graph:

• The function decreases from $x = -6$ to $x = -4$.
• It also decreases from $x = 0$ to $x = 2$.

Thus, the function is decreasing on the intervals $(-6, -4)$ and $(0, 2)$.

Summary:

• The function has a minimum of $-4$ at $x = -4$.
• The function is increasing on the interval $(-4, 0)$.
• The function is decreasing on the intervals $(-6, -4)$ and $(0, 2)$.

Would you like more details or have any questions about this? Here are some related questions you might want to explore:

1. How do you determine where a function is increasing or decreasing from a graph?
2. What does the derivative tell you about increasing and decreasing functions?
3. How do you find critical points of a function algebraically?
4. Can a function have more than one minimum or maximum point?
5. How would you express the behavior of the function as $x$ approaches infinity?

Tip: When analyzing a graph, always look at the direction of the slope to determine increasing and decreasing intervals.