Problem 68 - Algebra 2

Question

Identify each function as exponential growth or decay, and find the growth or decay factor. \(y=3 \cdot\left(\frac{1}{2}\right)^{x}\)

Step-by-Step Solution

Answer

The function \(y=3 \cdot\left(\frac{1}{2}\right)^{x}\) represents exponential decay and its decay factor is \(\frac{1}{2}\) .

Step 1: Identify the exponential function

The given function is \(y=3 \cdot\left(\frac{1}{2}\right)^{x}\) . This is already in the form of an exponential function \(y=ab^{x}\) , where \(a\) is the initial value and \(b\) is the base.

Step 2: Determine if it’s growth or decay

The base \(b\) in the function is \(\frac{1}{2}\) , which is less than 1. This indicates that we have exponential decay.

Step 3: Find the decay factor

In an exponential decay function, the decay factor is the base \(b\) . Therefore, the decay factor is \(\frac{1}{2}\) .

Key Concepts

Exponential Decay

When discussing exponential functions, one of the primary concepts is exponential decay. This occurs when a quantity decreases over time at a rate proportional to its current value. Imagine you have an ice cream cone on a hot day. It will melt faster at first and then more slowly as time passes, which is similar to exponential decay. The nature of exponential decay contrasts with exponential growth, where quantities increase over time (like collecting interest in a bank account). Decay happens when the base of an exponential function is less than one. In mathematical terms, for a function in the form of \( y = ab^x \), if \( 0 < b < 1 \), we say the function describes exponential decay.This concept frequently appears in real-world scenarios, such as radioactive decay, population decline, or cooling of an object. Understanding exponential decay helps in predicting how quickly a quantity reduces, allowing for informed decision-making in fields like science, engineering, and finance.

Decay Factor

The decay factor plays a crucial role in understanding exponential decay. It is essentially the base \( b \) of the exponential function \( y = ab^x \). In layman's terms, it tells us the fraction by which the original amount will decrease in each period. For example, if your decay factor is \( \frac{1}{2} \), this means you lose half of your substance each time period. The decay factor is less than one in all exponential decay scenarios, which ensures that the function’s output keeps reducing as \( x \) (often representing time) increases. Calculating the decay factor can assist in solving many practical problems such as finding how fast a drug leaves the body or estimating the depreciation of an asset. It provides a simple yet powerful tool to model decreasing phenomena effectively.

Identifying Functions

In mathematics, identifying whether a function exhibits exponential growth or decay is vital. Starting with a general exponential function expressed as \( y = ab^x \), where \( a \) is the starting value and \( b \) is the base, one determines the behavior by evaluating \( b \):- If \( b > 1 \), the function models exponential growth.- If \( 0 < b < 1 \), it models exponential decay.This clarity allows one to predict the pattern of change and adapt strategies in practical applications accordingly. To identify a function correctly:- Ensure your function is in the form \( ab^x \).- Examine the base \( b \) to ascertain whether it is greater than or less than one.- Recognize that a smaller base (between 0 and 1) leads to decay, while a larger base (greater than 1) signifies growth.By grasping this fundamental concept, students and professionals alike can better understand the dynamics of changing quantities across various domains.