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Understanding the rate of change in any context is crucial, as it measures how one quantity changes in relation to another. In our snowstorm example, we are given a situation with varying rates of snowfall over different periods, which is perfect for illustrating different rates of change within a single event. A rate of change is often expressed as a ratio or a fraction. In this case, it's inches of snow per hour.

To analyze, we examine the snowfall hour by hour. During the first two hours, the rate of change is constant at 1 inch per hour. We can denote this as \(1 \frac{inch}{hour}\). For the next six hours, it increases to 2 inches per hour or \(2 \frac{inches}{hour}\), and for the final hour, it decreases to 0.5 inch per hour or \(0.5 \frac{inch}{hour}\).

These different rates provide a vivid scenario to visualize the concept of rate of change as the slope of a graph in a piecewise-linear function, leading to a step-like construction where each 'step' corresponds to a constant rate of change for a given time interval.

Evaluating a function means finding the value of the function for a specific input. When we are given a piecewise-defined function such as the snowstorm function, \(f(t)\), it's important to apply the correct formula during function evaluation based on the interval that contains the input value.

For instance, if we were to evaluate the snow depth at the end of the second hour, we would use the first part of the function, \(1t\), and plug in \(2\) hours, yielding \( f(2) = 1\times2 = 2 \) inches of snow. Similarly, function evaluation at the end of the storm involves computing the snow depth at each phase change and summing up the results. This process shows the accumulated effect of different rates of change over time, demonstrating how piecewise functions can compartmentalize different behaviors of a system for practical problem-solving.

Visualizing piecewise functions can be incredibly informative, as it allows you to see the different rates and changes over time depicted graphically. Graphing our snowstorm's piecewise function, \(f(t)\), involves plotting the depth of snow over time, with the function changing its rule with each time interval.

A straightforward approach is to draw each segment of the graph within the intervals defined by the function. From \(0 < t \leq 2\), we draw a line with slope \(1\), from \(2 < t \leq 8\), a steeper line with slope \(2\), and from \(8 < t \leq 9\), a less steep line with slope \(0.5\). It's important to consider the starting point of each segment which can be calculated using the function. These joins in the graph correspond to the changes in the snowfall rate, providing a piecewise linear graph that is a series of straight lines with different slopes.

To make sure our graph is accurate, we use open and closed circles to denote inclusive and exclusive bounds of the intervals. In this case, a closed circle at \(t=2\) and \(t=8\) but an open circle at \(t=0\) and \(t=9\) would complete the graphical representation. Graphing this way not only helps you understand the transitions between different rates but also aids in visualizing the total accumulation of snow, as the final height of the last segment represents the total snow depth.