Problem 45 - Algebra 2

Question

Without graphing, determine whether each equation represents exponential growth or exponential decay. $$ y=\frac{7}{5}\left(\frac{e}{2}\right)^{x} $$

Step-by-Step Solution

Answer

The given function represents exponential growth.

Step 1: Identify the Base of the Exponential Function

In the function y = (7/5) * (e/2)^x, e/2 is the base. 'e' is Euler's number which approximately equals 2.718281828459. Hence, e/2 is equal to 2.71828/2 = 1.359140914 which is greater than 1.

Step 2: Compare the Base with 1

After finding the base value as 1.359140914, compare this with 1. Since 1.359140914 > 1, it can be determined that the exponential function represented by y = (7/5) * (e/2)^x is an example of exponential growth.

Key Concepts

Understanding Exponential Function

An exponential function is a specific type of mathematical expression where one of the variables appears as an exponent. It is typically written in the form \( y = a \cdot b^{x} \), where:

• \( y \) is the output of the function,
• \( a \) is a constant and represents the initial value or starting point,
• \( b \) is the base of the exponential function, and
• \( x \) is a variable, which usually represents time or another changing quantity.

Exponential functions are unique because they describe continuous and rapid growth or decay in various processes. They are widely used in fields such as biology, finance, and physics. When the base \( b \) is greater than 1, the function models exponential growth. Conversely, when \( 0 < b < 1 \), it models exponential decay. This differentiation is vital in determining whether a situation results in growth or reduction over time.
In the exercise, \( y = \left( \frac{7}{5} \right) \left(\frac{e}{2}\right)^{x} \), the expression describes how \( y \) changes as \( x \) varies, specifically indicating exponential growth due to its base being greater than 1.

What is the Base of an Exponential?

The base in an exponential function is the factor that an output is repeatedly multiplied by, as the input variable changes. In an exponential form \( y = a \cdot b^x \), \( b \) is the base. The base is crucial as it determines the nature of the growth or decay in the function.
When evaluating an exponential function, always identify the base first to understand the behavior of the function:

• If the base is greater than 1, it indicates exponential growth. The value of \( y \) increases as \( x \) increases.
• If the base is between 0 and 1, it signifies exponential decay. In this case, \( y \) decreases as \( x \) increases.
• If the base is exactly 1, the function remains constant since multiplying by 1 does not change the value.

In the given function \( y = \left(\frac{7}{5}\right) \left(\frac{e}{2}\right)^x \), the base is \( \frac{e}{2} \). Here, \( \frac{e}{2} \) equals approximately 1.359, indicating exponential growth because the base is greater than 1, leading to larger values of \( y \) as \( x \) increases.

Exploring Euler's Number

Euler's number, denoted as \( e \), is an important constant in mathematics, approximately equal to 2.71828. It's a fundamental constant that serves particular purposes in calculus and complex analysis. Euler's number is the base of natural logarithms and is extensively used in exponential growth models.
Properties of \( e \):

• \( e \) is an irrational number, meaning it cannot be expressed as a simple fraction. Its decimal representation goes on indefinitely without repeating.
• \( e \) is the unique number such that the derivative of the function \( f(x) = e^{x} \) is \( f'(x) = e^{x} \). This makes \( e \) extremely useful in calculus, especially for functions involving growth rates.
• Natural exponential functions, where \( e \) is the base, commonly model continuous growth processes found in nature, such as populations and radioactive decay.

In the provided example \( y = \left(\frac{7}{5}\right) \left(\frac{e}{2}\right)^x \), Euler’s number \( e \) is halved to form the base \( \frac{e}{2} \). Understanding \( e \) and its properties helps to interpret the exponential nature of various phenomena accurately.