Let's analyze the graph based on the questions you've posed:

1. Does the graph represent a function?

   • Yes, the graph represents a function. In a function, each input (time) must correspond to exactly one output (distance). In this graph, for every point along the horizontal axis (time), there is only one corresponding point on the vertical axis (distance). Therefore, it passes the "vertical line test," indicating that it is indeed a function.

2. What is the domain?

   • The domain of a function is the set of all possible inputs, in this case, all values of time for which there is a corresponding distance. Since the graph starts from a certain point on the left and ends on the right, the domain would be the interval covering all time values from the starting to the ending point.

3. What is the range?

   • The range is the set of all possible output values, meaning all values of distance that the graph reaches. Observing the graph, the range would be the interval from the lowest point (minimum distance) to the highest point (maximum distance) on the vertical axis.

4. What is the maximum?

   • The maximum is the highest point on the graph, which represents the greatest distance reached over the given time period. You can identify this by finding the peak point in the upward direction.

5. What is the minimum?

   • The minimum is the lowest point on the graph, representing the smallest distance reached (likely at the start if it begins at zero). This is the lowest point along the vertical axis that the graph reaches.

6. Identify the intervals where the graph is increasing. What does this mean in the context of the problem?

   • The intervals where the graph is increasing are the segments where the graph moves upward as time progresses. This means that the distance is increasing over these intervals, indicating that movement away from the starting point is occurring, or the object is moving farther from its initial position.

7. Identify the intervals where the graph is decreasing. What does this mean in the context of the problem?

   • The intervals where the graph is decreasing are the segments where the graph moves downward as time progresses. This indicates that the distance is decreasing, meaning the object is moving closer to the starting point or returning back toward an earlier position.

8. Identify the intervals where the graph is constant. What does this mean in the context of the problem?

   • The intervals where the graph is constant are the horizontal segments where the distance does not change over time. This means that the object is stationary during these intervals, remaining at a constant distance without moving forward or backward.

Would you like a more detailed breakdown or a closer look at specific points on the graph?

Further Exploration Questions:

1. What could cause an object to have increasing, decreasing, and constant distances over time?
2. How would you calculate the average speed in intervals where the graph is increasing?
3. What type of real-life situation could this graph represent?
4. How could we determine the total distance traveled if the object changes direction multiple times?
5. If given numerical values, how would you determine the exact maximum and minimum values?

Tip: In distance-time graphs, the slope represents speed, with steeper slopes indicating faster speeds.