Problem 25 - Algebra 2

Question

Determine whether each function represents exponential growth or decay. $$ y=3\left(\frac{5}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The function represents exponential growth.

Step 1: Identifying Base of Exponent

In the function \( y = 3\left(\frac{5}{2}\right)^{x} \), identify the base of the exponent. Here, the base is \( \frac{5}{2} \).

Step 2: Classifying the Base

Determine whether the base \( \frac{5}{2} \) is greater than 1 or between 0 and 1. Since \( \frac{5}{2} = 2.5 \), which is greater than 1, we identify the base as greater than 1.

Step 3: Conclusion on Growth or Decay

If the base is greater than 1, the function represents exponential growth. Since \( \frac{5}{2} = 2.5 \) is greater than 1, the function \( y = 3\left(\frac{5}{2}\right)^{x} \) represents exponential growth.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that model growth or decay in a way that the rate of change is proportional to the current value. They are commonly used in various areas such as biology, finance, and physics. The generic form of an exponential function is \(y = ab^x\), where:

• \(a\) is a constant coefficient.
• \(b\) is the base of the exponent.
• \(x\) is the exponent variable, typically representing time or a sequence parameter.

Exponential growth occurs when the base \(b\) is greater than 1, showing how quantities rapidly increase over time. Conversely, when \(0 < b < 1\), it represents exponential decay, where values decrease over time. The beauty of these functions lies in their ability to model real-world processes, like population growth or radioactive decay, enabling predictions and deeper understanding of change dynamics.

Base of an Exponent

The base of an exponent in an exponential function determines the nature of the growth or decay of the function. In the expression \(y = ab^x\):

• \(b\) is the crucial element that dictates the function's behavior.
• If \(b > 1\), the function represents exponential growth.
• If \(0 < b < 1\), it symbolizes exponential decay.

Recognizing the base is essential when analyzing exponential functions. For example, consider the base \(\frac{5}{2}\) from the function \(y = 3\left(\frac{5}{2}\right)^{x}\). Since \(\frac{5}{2} = 2.5\), which is greater than 1, this base characterizes a function as experiencing exponential growth. Understanding how the base influences the entire expression is key to comprehending the function's impact over time.

Mathematical Function Analysis

Mathematical function analysis involves breaking down functions to understand their properties and behavior. In the context of exponential functions, this analysis helps determine whether the function models growth or decay. Here's how you can analyze functions like \(y = 3\left(\frac{5}{2}\right)^{x}\):

• Identify the components of the function: coefficient, base, and exponent.
• Examine the base to determine if it's greater than 1 or between 0 and 1.
• Conclude whether the function represents growth (base greater than 1) or decay (base between 0 and 1).

Through function analysis, not only do you classify the function type, but you can also predict behavior over time. Such insights are invaluable for making informed decisions in fields that rely heavily on data and predictions, such as finance and environmental science.