In Exercises 79-82, sketch and label the graph of a function...

In Exercises $79-82,$ sketch and label the graph of a function that describes the given situation. Suppose that after you drink a cup of coffee the amount of caffeine in your body rises sharply and then decreases by half every hour. You have one cup of coffee in the morning and then no more.

In Exercises $79-82,$ sketch and label the graph of a function that describes the given situation.
Suppose that after you drink a cup of coffee the amount of caffeine in your body rises sharply and then decreases by half every hour. You have one cup of coffee in the morning and then no more.

Key Concepts

Graphing Functions

Graphing functions involves plotting the relationship between the independent variable (time) and the dependent variable (amount of caffeine) on a coordinate plane. This concept includes labeling the axes, indicating key points such as the initial value and successive points of halved caffeine levels, and showing the asymptotic behavior as the caffeine level approaches zero over time.

Exponential Functions

Exponential functions are mathematical functions that model exponential growth or decay processes. They are particularly relevant when describing how certain quantities change rapidly at first and then level off. In this case, an exponential decay function can be used to represent the decreasing concentration of caffeine after coffee consumption.

Exponential Decay

Exponential decay describes a process whereby a quantity decreases by a consistent percentage over equal time intervals. In this context, it means that after reaching a peak, the level of caffeine in the body decreases by a fixed fraction each hour, which is a signature behavior of exponential decay.

Half-Life

Half-life is the time it takes for a quantity undergoing exponential decay to reduce to half its initial value. Here, the concept is used to show that every hour, the amount of caffeine in the body is halved, making it a critical parameter for modeling how caffeine is metabolized over time.

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