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When discussing the motion of an object, such as a car on a journey, a crucial concept is that of average velocity. Average velocity gives a simple overview of how fast something moves over a certain period of time. To calculate average velocity, you divide the total distance traveled by the total time it took to travel that distance. Here's the formula:

• Average Velocity: \( \text{Average Velocity} = \frac{\text{Total Distance}}{\text{Total Time}} \)

In the context of the given problem, the car travels a total of 200 miles in 4 hours. So, the average velocity is \( \frac{200\text{ miles}}{4\text{ hours}} = 50\text{ mph} \). This means that overall, the car's speed is 50 miles per hour on average, which summarizes the entire trip without focusing on variations that might occur throughout the journey. It offers a straightforward measure of the car's performance over the trip.

In this scenario, because the car travels at a constant speed (as indicated by the average velocity calculation), the velocity function can be classified as a constant function. A constant function is one where the output value is the same no matter what the input value is.

• For this problem, the velocity of the car, \( v(t) = 50 \text{ mph} \), remains constant for any time \( t \) within the 4-hour journey.
• This means that at any point during the trip, the car maintains a speed of 50 mph.

Understanding constant functions is vital in mathematics because they simplify calculations. In driving scenarios like this one, a constant speed over a period indicates no acceleration or deceleration – the car just keeps moving at the same pace, making predictions and planning straightforward. This helps us relate mathematical functions to real-world behavior in a direct and practical way.

Graphing the velocity function provides a visual representation of the car's motion over time. On a graph, the x-axis represents time and the y-axis reflects velocity.

• In this problem, you would draw a horizontal line on the graph at \( y = 50 \) mph.
• This line extends from \( x = 0 \) hours to \( x = 4 \) hours.

This horizontal line illustrates that the velocity does not change over time, confirming it is a constant function. Each point on the line shows that, at any given moment between 0 and 4 hours, the velocity is consistently 50 mph. Graphs like this help to easily visualize and understand the behavior of functions. They demonstrate how constant functions maintain stability, making the link between abstract mathematical concepts and their real-world applications crystal clear.