The graph shows the depth of water, in feet, of a kids' swimming pool over a certain amount of time, in minutes.

graph with the title kids' swimming pool with the x axis labeled time in minutes and the y axis labeled depth in feet, with a line segment from 0 comma 0 to 40 comma 4, a line segment from 40 comma 4 to 60 comma 4, a line segment from 60 comma 4 to 100 comma 3, a line segment from 100 comma 3 to 140 comma 3, and another line segment from 140 comma 3 to 180 comma 0

Part A: Identify the interval(s) where the graph is increasing, decreasing, and constant. (3 points)

Part B: What are the domain and range of the relation? (3 points)

Part C: Does the graph represent a function? Explain your reasoning. (3 points)

Part D: Write a detailed description of the graph using the real-world scenario and the information from Part A. (3 points)

Let's analyze the graph of the kids' swimming pool and answer the questions step by step.

### Part A: Identify the interval(s) where the graph is increasing, decreasing, and constant.

1. **Increasing:**
   - From \( t = 0 \) to \( t = 40 \) minutes, the depth increases from 0 feet to 4 feet.
   - Interval: \( [0, 40] \)

2. **Constant:**
   - From \( t = 40 \) to \( t = 60 \) minutes, the depth remains constant at 4 feet.
   - From \( t = 100 \) to \( t = 140 \) minutes, the depth remains constant at 3 feet.
   - Intervals: \( [40, 60] \) and \( [100, 140] \)

3. **Decreasing:**
   - From \( t = 60 \) to \( t = 100 \) minutes, the depth decreases from 4 feet to 3 feet.
   - From \( t = 140 \) to \( t = 180 \) minutes, the depth decreases from 3 feet to 0 feet.
   - Intervals: \( [60, 100] \) and \( [140, 180] \)

### Part B: What are the domain and range of the relation?

1. **Domain:**
   - The domain is the set of all possible time values for which the depth is defined.
   - Domain: \( [0, 180] \) minutes.

2. **Range:**
   - The range is the set of all possible depth values.
   - Range: \( [0, 4] \) feet.

### Part C: Does the graph represent a function? Explain your reasoning.

Yes, the graph represents a function. In mathematical terms, a relation is a function if each input (time in minutes) corresponds to exactly one output (depth in feet). In this case, for each value of time between 0 and 180 minutes, there is a unique depth value. Therefore, the graph is a function.

### Part D: Write a detailed description of the graph using the real-world scenario and the information from Part A.

The graph titled "Kids' Swimming Pool" illustrates the depth of water in a swimming pool over time. Initially, from 0 to 40 minutes, the depth of the pool increases steadily from 0 feet to 4 feet, indicating the pool is being filled with water. For the next 20 minutes, from 40 to 60 minutes, the depth remains constant at 4 feet, suggesting that the pool is fully filled and no additional water is being added.

From 60 to 100 minutes, the depth decreases gradually from 4 feet to 3 feet, indicating some water is being drained or splashed out. The depth then stays constant at 3 feet from 100 to 140 minutes, implying no significant changes in the water level. Finally, from 140 to 180 minutes, the depth decreases again from 3 feet to 0 feet, showing the pool is being drained completely. The domain of the graph is from 0 to 180 minutes, and the range is from 0 to 4 feet, confirming the pool's depth varies within these limits.

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Would you like more details or have any questions? Here are some questions you might find helpful:

1. How do you determine the intervals of increase, decrease, and constancy in a graph?
2. What methods can be used to find the domain and range of a function?
3. Why is it important to verify if a graph represents a function?
4. How can you interpret real-world scenarios from graphical data?
5. What are some common features to look for in a piecewise linear graph?

**Tip:** When analyzing graphs, always pay close attention to the axes labels and the units of measurement, as they provide critical context for interpreting the data accurately.

Explore how to analyze a graph depicting the depth of a kids' swimming pool over time. Identify intervals of increase, decrease, and constancy, and determine the domain and range of the relation. Understand whether the graph represents a function and interpret the real-world scenario.

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