Problem 44 - Algebra 2

Question

Without graphing, determine whether each equation represents exponential growth or exponential decay. $$ y=\left(\frac{1}{e}\right)^{x} $$

Step-by-Step Solution

Answer

The given equation \(y= (\frac{1}{e})^x\) represents exponential decay.

Step 1: Identify Exponential Function Base

Identify the base of exponential function. In this particular function \(y= (\frac{1}{e})^x\), the base is \(\frac{1}{e}\).

Step 2: Compare Base to 1

Next, compare base to 1. Notice that as \(e ≈ 2.718\), the base \(\frac{1}{e} ≈ 0.36788\), which is less than 1.

Step 3: Determine Exponential Growth or Decay

Recall the rules: If \(a>1\), the function represents exponential growth, and if \(0

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept that describes a process where quantities increase rapidly over time. This type of growth is characterized by its increasing rate, meaning the more time passes, the faster the growth.Understanding exponential growth can be made easier by looking at its mathematical expression:- An exponential function in the form of \(y = a^x\) is said to exhibit exponential growth if the base \(a\) is greater than 1.With exponential growth:- The variable grows by a constant percentage or factor over equal increments of time.- This pattern results in a steep upward curve when graphed.Real-world examples include:- Population growth where a large number of offspring leads to an ever-increasing population.- Compound interest in finance, where money grows because interest is earned on previously accumulated interest.Exponential growth is pervasive in many fields of science, economics, and technology, making understanding its principles crucial.

Exponential Decay

Exponential decay is the counterpoint to exponential growth, describing a process where quantities decrease at a consistently reducing rate over time.Let's break it down:- An exponential function written as \(y = a^x\) demonstrates exponential decay if the base \(a\) is between 0 and 1.Key characteristics of exponential decay include:- The rate of change is proportional to the current value, resulting in a quick reduction initially which slows down over time.- Graphically, this manifests as a curve that falls steeply at first before leveling off.Common examples of exponential decay:- Radioactive decay where unstable atoms lose particles at a predictable rate.- Depreciation of assets like cars which lose value progressively over their lifetime.Understanding exponential decay is essential in fields such as physics, economics, and environmental science, guiding us in predicting how things change with time.

Base of Exponential Function

The base of an exponential function is a critical component that dictates how the function behaves, shaping whether it will experience growth or decay.To comprehend this better:- The general form \(y = a^x\) defines \(a\) as the base which determines the rate and nature of the function's change.Key points related to the base include:- If the base \(a\) is greater than 1, the function exhibits growth, implying the output expands over time.- Conversely, if \(0 < a < 1\), it signals decay, meaning the output diminishes as time progresses.In practical terms:- The base influences the steepness and direction of the curve on a graph, directly impacting decision-making in applications ranging from finance to sciences.- For instance, in our given equation \(y = (\frac{1}{e})^x\), with \(e\) approximately equal to 2.718, hence \(\frac{1}{e} \approx 0.36788\), it leads to exponential decay due to the base being less than 1.Mastering the concept of the base in exponential functions allows for a deeper understanding of how natural processes evolve and change over time.

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Key Concepts

Exponential Growth

Exponential growth is a fascinating concept that describes a process where quantities increase rapidly over time. This type of growth is characterized by its increasing rate, meaning the more time passes, the faster the growth.Understanding exponential growth can be made easier by looking at its mathematical expression:- An exponential function in the form of \(y = a^x\) is said to exhibit exponential growth if the base \(a\) is greater than 1.With exponential growth:- The variable grows by a constant percentage or factor over equal increments of time.- This pattern results in a steep upward curve when graphed.Real-world examples include:- Population growth where a large number of offspring leads to an ever-increasing population.- Compound interest in finance, where money grows because interest is earned on previously accumulated interest.Exponential growth is pervasive in many fields of science, economics, and technology, making understanding its principles crucial.

Exponential Decay

Exponential decay is the counterpoint to exponential growth, describing a process where quantities decrease at a consistently reducing rate over time.Let's break it down:- An exponential function written as \(y = a^x\) demonstrates exponential decay if the base \(a\) is between 0 and 1.Key characteristics of exponential decay include:- The rate of change is proportional to the current value, resulting in a quick reduction initially which slows down over time.- Graphically, this manifests as a curve that falls steeply at first before leveling off.Common examples of exponential decay:- Radioactive decay where unstable atoms lose particles at a predictable rate.- Depreciation of assets like cars which lose value progressively over their lifetime.Understanding exponential decay is essential in fields such as physics, economics, and environmental science, guiding us in predicting how things change with time.

Base of Exponential Function

The base of an exponential function is a critical component that dictates how the function behaves, shaping whether it will experience growth or decay.To comprehend this better:- The general form \(y = a^x\) defines \(a\) as the base which determines the rate and nature of the function's change.Key points related to the base include:- If the base \(a\) is greater than 1, the function exhibits growth, implying the output expands over time.- Conversely, if \(0 < a < 1\), it signals decay, meaning the output diminishes as time progresses.In practical terms:- The base influences the steepness and direction of the curve on a graph, directly impacting decision-making in applications ranging from finance to sciences.- For instance, in our given equation \(y = (\frac{1}{e})^x\), with \(e\) approximately equal to 2.718, hence \(\frac{1}{e} \approx 0.36788\), it leads to exponential decay due to the base being less than 1.Mastering the concept of the base in exponential functions allows for a deeper understanding of how natural processes evolve and change over time.