Problem 61 - Algebra 2

Question

OPEN ENDED Give an example of a value of \(b\) for which \(y=b^{x}\) represents exponential decay.

Step-by-Step Solution

Answer

An example is \(b = 0.5\), which causes exponential decay.

Step 1: Understanding Exponential Decay

Exponential decay occurs when a quantity decreases over time. In exponential functions like \(y = b^x\), the base \(b\) determines whether the function represents growth or decay. For exponential decay, the base \(b\) must be between 0 and 1, i.e., \(0 < b < 1\).

Step 2: Choosing an Example Value

To provide an example of a base that leads to exponential decay, we need a value of \(b\) such that \(0 < b < 1\). Let's choose \(b = 0.5\).

Step 3: Check the Properties

The chosen value \(b = 0.5\) is indeed between 0 and 1. Therefore, \(y = 0.5^x\) describes a situation where the value of \(y\) decreases as \(x\) increases, confirming that this is an example of exponential decay.

Key Concepts

Base of an Exponential Function

The base of an exponential function is a fundamental component that directly influences the nature of the function. In any exponential function, typically written as \( y = b^x \), \( b \) is the base. This base dictates how the values of \( y \) change as \( x \) changes. The characteristics of the base determine whether the function models growth or decay.

• **If \( b > 1 \)**: The function models exponential growth. As \( x \) increases, \( y \) grows larger.
• **If \( 0 < b < 1 \)**: The function models exponential decay. As \( x \) increases, \( y \) decreases.

The base does not only tell us whether the function increases or decreases, but also the rate at which these changes happen. For instance, a base \( b = 0.5 \) indicates that \( y \) halves with every increase in \( x \).
Understanding the role of the base helps in predicting and analyzing the behavior of exponential functions in real-world scenarios, such as population growth or radioactive decay.

Exponential Functions

Exponential functions describe relationships where a constant percentage rate of change applies to any size of quantity—in other words, the change is proportional to the current value. One of the most common ways this is expressed is with the function form \( y = b^x \). The variable \( x \) usually represents time, whereas \( y \) represents the quantity of interest.

• Exponential functions are widely used in fields like science, economics, and engineering.
• They allow us to model processes such as population growth, radioactive decay, and interest compounding.

Exponential growth or decay can be visually represented on a graph. These graphs are characterized by either rapid increase or decrease, depending respectively on whether \( b > 1 \) or \( 0 < b < 1 \). The key aspect is the nature of the change: it is not linear, but rather exponential, resulting in more dramatic effects as time progresses.
Being able to recognize and understand exponential functions is crucial because they appear in many real-world contexts.

Value of b Between 0 and 1

When the base \( b \) of an exponential function is between 0 and 1, the function will demonstrate exponential decay. This means the function will decrease over time as the input—usually time or another variable—grows. This key property distinguishes exponential decay from exponential growth.
Choosing a value for \( b \) such as 0.5 illustrates exponential decay. In this case, for \( y = 0.5^x \):

• Each time \( x \) increases by 1, the value of \( y \) is halved.
• This demonstrates a rapid decline as \( x \) becomes larger.

Exponential decay is frequently observed in phenomena like radioactive decay, where the amount of a substance decreases over time as it breaks down. It's crucial to note that for decay, \( b \) must satisfy \( 0 < b < 1 \), meaning it is a fraction or decimal less than 1.
Mastering this concept means understanding not just the mathematical function, but also being able to spot real-life applications. Recognizing an exponential decay scenario is vital in fields such as chemistry, finance, and environmental science.