Decay Function

Definition

A decay function is a type of exponential function that models the gradual decrease or diminishing of a quantity over time. It is characterized by an initial value that decreases at a constant rate, often used to represent the behavior of radioactive materials, population decline, and other natural phenomena.

5 Must Know Facts For Your Next Test

1. Decay functions are a specific type of exponential function where the base $b$ is less than 1, resulting in a decreasing or decaying trend over time.
2. The general form of a decay function is $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the decay factor (a value between 0 and 1).
3. The rate of decay is determined by the value of the decay factor $b$, with smaller values of $b$ indicating a faster rate of decay.
4. Decay functions often model the behavior of radioactive materials, where the half-life represents the time it takes for the quantity to decrease to half of its initial value.
5. The graph of a decay function is characterized by a decreasing, concave-down curve that approaches a horizontal asymptote as the independent variable (often time) increases.

Review Questions

• Explain how the decay factor $b$ in a decay function affects the rate of decay.
  • The decay factor $b$ in a decay function determines the rate of decay. When $b$ is a value between 0 and 1, it represents a decreasing exponential function. The smaller the value of $b$, the faster the rate of decay. For example, if $b = 0.5$, the quantity will decrease by 50% every unit of time, whereas if $b = 0.9$, the quantity will decrease more slowly, by 10% every unit of time. The decay factor directly influences the steepness of the decay curve and the time it takes for the function to approach its horizontal asymptote.
• Describe the relationship between the half-life of a decay function and the decay factor $b$.
  • The half-life of a decay function is the time it takes for the quantity to decrease to half of its initial value. The half-life is inversely related to the decay factor $b$. Specifically, the half-life can be calculated as $t_{1/2} = \frac{\ln(2)}{\ln(b)}$, where $\ln$ represents the natural logarithm. This means that as the decay factor $b$ decreases (indicating a faster rate of decay), the half-life also decreases, and vice versa. Understanding the relationship between the half-life and the decay factor is crucial for modeling and analyzing decay processes in various applications.
• Explain how the graph of a decay function relates to the concept of an asymptote.
  • The graph of a decay function is characterized by a decreasing, concave-down curve that approaches a horizontal asymptote as the independent variable (often time) increases. This horizontal asymptote represents the eventual limit or minimum value that the function approaches, but never actually reaches. The closer the decay factor $b$ is to 0, the faster the function will approach its asymptote. Understanding the behavior of decay functions and their asymptotic properties is essential for accurately modeling and predicting the long-term behavior of systems that exhibit exponential decay, such as radioactive decay, population decline, and various other natural and technological processes.

Related terms

Exponential Function:

A function that grows or decays at a constant rate, expressed in the form $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the base or growth/decay factor.

Half-Life: The time it takes for a quantity to decrease to half of its initial value in a decay function.

Asymptote:

A line that a curve approaches but never touches, in the case of a decay function, the horizontal asymptote represents the eventual limit or minimum value the function approaches.