Let’s analyze the graph and question provided. The function $C(t)$ models the percentage of charge in a phone battery over time $t$. The task is to determine which statement best describes the behavior of the graph.

Observations from the Graph:

1. The graph shows a steady decrease in the battery percentage as time progresses.
2. The rate of decrease (slope) varies over time, as the curve is not linear—it becomes less steep in some parts and more steep in others.

Explanation of Each Option:

• A: Incorrect. The graph does not show any instances of increase; it only decreases.
• B: Incorrect. While it correctly mentions the charge always decreases, it inaccurately claims the function is "sometimes decreasing and sometimes increasing."
• C: Incorrect. This option states that the decrease happens "at the same rate at all times," which is incorrect since the graph’s slope varies.
• D: Correct. The function is always decreasing, and the rate of decrease varies over time, as seen by the non-linear nature of the graph.

Final Answer:

D is the accurate description of the function and its behavior.

Let me know if you want detailed reasoning or examples!

Related Questions:

1. What is the mathematical significance of the slope of the graph at any point?
2. How does a non-linear function differ from a linear function in this context?
3. What does a steep slope indicate in terms of battery charge usage?
4. Can you think of real-life factors that could explain why the battery charge decreases at varying rates?
5. How would the graph look if the phone were plugged in to charge periodically?

Tip:

When interpreting graphs, always analyze the direction (increasing or decreasing) and the shape (linear or curved) to understand the behavior of the function.