1. The problem is to analyze the height of a hot air balloon over time given the data points: (1,8), (2,11), (3,9), and (4,12). 2. We want to understand the relationship between time (minutes) and height (meters) and describe the graph properly. 3. The data points are: - At time $t=1$ minute, height $h=8$ m - At time $t=2$ minutes, height $h=11$ m - At time $t=3$ minutes, height $h=9$ m - At time $t=4$ minutes, height $h=12$ m 4. The graph is a line graph connecting these points in order, showing how height changes over time. 5. Important notes: - The line does not start at the origin $(0,0)$ because the first time measurement is at 1 minute. - The vertical axis represents height in meters, ranging from 0 to at least 20 meters to accommodate the data and future values. - The horizontal axis represents time in minutes. - The axes labels should be swapped if needed, but here time is horizontal and height is vertical, which is standard. - A key should be included to indicate what the line represents (height over time). 6. To describe the graph functionally, we can consider the points as discrete data rather than a continuous function, but a piecewise linear function $h(t)$ can be defined connecting these points. 7. The graph shows fluctuations in height: it rises from 8 to 11 meters between 1 and 2 minutes, drops to 9 meters at 3 minutes, then rises again to 12 meters at 4 minutes. 8. This analysis helps understand the balloon's height changes over the given time interval. Final answer: The height of the hot air balloon over time is represented by the points $(1,8)$, $(2,11)$, $(3,9)$, and $(4,12)$ connected by straight lines, with height on the vertical axis (0 to 20 meters) and time on the horizontal axis (minutes). The line does not start at the origin, and the graph includes appropriate axis labels and a key.