Problem 46 - Algebra 2

Question

Without graphing, determine whether each equation represents exponential growth or exponential decay. $$ f(x)=\left(\frac{e}{3.7}\right)^{x} $$

Step-by-Step Solution

Answer

The given function represents exponential decay.

Step 1: Evaluate the Base

Evaluate the base by dividing \(e\) by 3.7. Since 3.7 is greater than \(e\) (approximately 2.71828), the base is less than 1. A quick calculation reveals that \(\frac{e}{3.7} \approx 0.7344\).

Step 2: Determine Growth or Decay

Since the base of the function, 0.7344, is between 0 and 1 (excluding 0 and 1 themselves), the given exponential function represents exponential decay.

Key Concepts

Exponential Growth

When discussing exponential growth, we're talking about a pattern where the value increases rapidly over time. This growth occurs when a quantity increases by a consistent percentage in a consistent time period. Exponential growth has a base greater than 1.

• Examples might include populations, investments, or viral videos being shared more and more quickly.
• With a base greater than 1, each step leads to a bigger escalation, like compounding interest.

It’s the kind of growth where things escalate swiftly in a multiplicative manner, making it a vital concept in fields like finance and biology. Understanding exponential growth helps in predicting future patterns and making crucial decisions. With rapid advancements in technology and global connectivity, recognizing how exponential growth operates can alert us to big changes coming fast.

Function Base

The function base in exponential functions is a crucial component. This base determines whether the exponential pattern will be growth or decay, impacting how we interpret changes over time.

• If the base is greater than 1, it signifies exponential growth.
• If the base is between 0 and 1, the function exhibits exponential decay.

In the original exercise, the base of the function was given as \( \left(\frac{e}{3.7}\right) \). Evaluating the base involves simplifying this expression. Since \( e \approx 2.71828 \), dividing \( e \) by 3.7 gives approximately 0.7344, which is under 1, flagging it as exponential decay.

Evaluating Expressions

Evaluating expressions is the process of finding the value of an expression for given values of the variables that occur in it. In the context of functions like those discussed in the exercise, this involves carrying out operations like division to discern the nature of the function.To evaluate the expression in the function \( f(x)=\left(\frac{e}{3.7}\right)^{x} \), we first determine the base by calculating \( \frac{e}{3.7} \), which yields approximately 0.7344.By evaluating expressions, we can determine essential characteristics of mathematical functions, leading to a better comprehension of how they behave over different variables and time. This skills set enhances problem-solving abilities and understanding of functional relationships.

Algebra 2

Algebra 2 is a foundational math course that often deals with functions, including exponential functions as seen in the exercise. It builds on concepts from Algebra 1 and dives deeper into more complex equations and functions. In Algebra 2, students explore:

• The behavior of different types of functions, including exponential, quadratic, polynomial, and more.
• Concepts such as the base of a function, transformations, and understanding growth vs. decay in natural contexts.

By mastering these topics, students develop strong analytical skills and the ability to tackle complex problems. Algebra 2 not only solidifies basic algebraic concepts but also introduces students to real-world applications through modeling and simulations, preparing them for higher education and various career paths.