Final answer:
The average rate of change in mathematics can be determined from graphs by examining the slope over given intervals. Different behaviors, such as a positive, negative, or zero rate, can be deduced from the way the graph behaves over these intervals. Collecting and analyzing data from surveys can provide the needed information to plot such graphs.
Step-by-step explanation:
The average rate of change refers to the change in value of a quantity divided by the elapsed time. To determine whether the average rate of change is positive, negative, or zero for a given interval, one must look at the behavior of the function or data within that interval. When we analyze graphs, such as velocity vs. time or acceleration vs. time graphs, the slope indicates the rate of change. A positive slope means a positive rate of change, a negative slope indicates a negative rate of change, and a horizontal line indicates a rate of change of zero.
For example, when a graph has a positive, constant slope, the rate of change is positive and constant. When the slope starts positive but decreases to zero, the rate of change is positive and decreasing. If the slope goes from zero to negative, then we have a rate of change that starts at zero and becomes more negative over time. Graphing distance vs. time can reveal such behaviors when you plot the average distance over average time and analyze the resulting line.
In collecting averages of data, such as the average changes of classmates, the process involves surveying, recording data, and then graphing or analyzing it to understand the patterns and determine rates of change.