Problem 17 - Algebra 2

Question

Without graphing, determine whether each function represents exponential growth or exponential decay. $$ f(x)=2(0.65)^{x} $$

Step-by-Step Solution

Answer

The given function \(f(x) = 2(0.65)^x\) represents exponential decay.

Step 1: Identify the base of the exponent

Looking at function \(f(x) = 2(0.65)^x\), it is in the form of \(a \cdot b^x\). Here, \(b = 0.65\) is the base of the exponent.

Step 2: Determine whether the base is less or more than 1

Compare the base of the exponent \(b=0.65\) with 1. It is observed that \(0.65 < 1\).

Step 3: Conclude if the function represents exponential growth or decay based on the comparison

Since the base of the exponent \(0.65\) is less than 1, it indicates that the function \(f(x) = 2(0.65)^x\) represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth refers to the increase of a quantity over time, where the rate of growth is proportional to the current amount. This creates a rapid increase as time progresses. It occurs when the base of the exponent in the function is greater than 1.
For instance, consider the function \( f(x) = a \, b^x \), where \( b > 1 \). Here, the base \( b \) directly determines the multiplier effect in each time period.

• A base of \( b = 2 \) means the quantity doubles in each step.
• A base of \( b = 1.5 \) indicates a 50% increment in each cycle.

In the real world, examples include populations growing without constraints, or compound interest in finance, where the growth occurs at an increasing rate due to the constant compounding factor.

Exponential Decay

Exponential decay is the process of reducing a quantity over time. It happens when the function's base of the exponent is between 0 and 1. This results in the quantity decreasing by a consistent percentage in each time period.
Consider the example \( f(x) = a \, b^x \), where \( 0 < b < 1 \). The base \( b \) dictates how much the quantity shrinks with each increment of \( x \).

• If \( b = 0.5 \), the quantity halves each time.
• With \( b = 0.9 \), there is a 10% reduction with each step.

A practical instance of exponential decay could be observed in the depreciation of a car's value or the decay of radioactive substances over time. These processes highlight the steady reduction as modeled by the exponential decay function.

Base of an Exponent

In the context of exponential functions, the base of an exponent is the constant that is raised to a variable exponent. It is a crucial part of determining the nature of the function—whether it depicts growth or decay. In a typical formulation \( a \, b^x \), the base \( b \) plays a pivotal role.
For any exponential function:

• If \( b > 1 \), it signifies exponential growth. The value of the function increases as the exponent rises.
• If \( 0 < b < 1 \), it indicates exponential decay. The function's value decreases as the exponent climbs.

Thus, the base fundamentally determines the behavior and direction of change of the function's output. Understanding the base helps us predict how a function will behave over different values of \( x \).