Based on the graph provided, let's analyze the behavior of the function:

Analysis of the Graph

1. Increasing Intervals: The function is increasing where the graph moves upwards as we move from left to right. This happens on the right side of the vertex.

   • In this graph, the function starts increasing from the vertex point, which is at $x = 1$, and continues increasing towards $x = 5$.

   • Thus, the interval where the function is increasing is $(1, 5)$.

2. Decreasing Intervals: The function is decreasing where the graph moves downwards as we move from left to right. This happens on the left side of the vertex.

   • In this graph, the function decreases from $x = -5$ to the vertex point $x = 1$.

   • Thus, the interval where the function is decreasing is $(-5, 1)$.

3. Constant Intervals: The function is constant if it maintains the same $y$-value over an interval. There is no section in this graph where the $y$-value remains constant over an interval.

   • Therefore, there are no intervals where the function is constant.

Conclusion

• For part (a), the function is increasing on the interval: $(1, 5)$.
• For part (b), the function is decreasing on the interval: $(-5, 1)$.
• For part (c), there are no constant intervals.

Let me know if you want further details or have any questions.

Follow-up Questions:

1. What is the vertex of this function and its significance?
2. How do you determine whether a function is increasing or decreasing from its derivative?
3. What is the range of this function based on the graph?
4. How does the symmetry of the graph indicate the nature of the function?
5. How would the intervals change if the function were shifted horizontally or vertically?

Tip: When determining intervals of increase or decrease, always focus on the behavior of the graph as you move from left to right.