Problem 22 - Algebra 2

Question

Without graphing, determine whether each function represents exponential growth or exponential decay. $$ y=\frac{1}{100}\left(\frac{4}{3}\right)^{x} $$

Step-by-Step Solution

Answer

The given function represents exponential growth.

Step 1: Identify the base of the exponent

In our given function \(y=\frac{1}{100}\left(\frac{4}{3}\right)^{x}\), the base \(b\) of the exponent is \(\frac{4}{3}\).

Step 2: Determine if the base is greater than or lesser than 1

If the base \(b\) of the exponent is greater than one, the function represents exponential growth. If the base of the exponent is between 0 and 1, then it indicates exponential decay. In our case, the base \(b\) is \(\frac{4}{3}\), which is larger than 1.

Step 3: Conclude the type of exponential function

Since the base \(b = \frac{4}{3} > 1\), the function represents exponential growth.

Key Concepts

Exponential Decay

Exponential decay is a process where quantities decrease at a rate proportional to their current value. It can be thought of as the opposite of exponential growth. The distinguishing feature of exponential decay is that the base of the exponent is between zero and one. This means that with each increase in the exponent, the resulting value gets smaller and smaller.

Imagine you have a function like \( y = a \cdot b^x \), where \( a \) is a constant and \( b \) is the base of the exponent. For exponential decay:

• The base \( b \, (0 < b < 1) \).
• This results in the value of \( y \) decreasing as \( x \) increases.

In the process of exponential decay, the function value approaches zero as the exponent increases, but never truly reaches it. This concept is often used to model real-world phenomena such as radioactive decay or depreciation of assets. Understanding exponential decay helps students visualize and predict how certain quantities diminish over time.

Exponential Functions

Exponential functions are mathematical expressions in the form \( y = a \cdot b^x \), where \( a \) and \( b \) are constants with \( b > 0 \) and \( b eq 1 \). These functions are powerful tools for modeling situations involving rapid change - both growth and decay.

For exponential functions, it's crucial to:

• Identify the base \( b \), which determines the function's behavior (growth if \( b > 1 \), decay if \( 0 < b < 1 \)).
• Recognize applications in real-life, such as population growth, interest calculations, or cooling temperatures.

Exponential functions, due to their nature, can describe phenomena that linear functions simply cannot. They encapsulate the idea of quantities increasing or decreasing multiplicatively, capturing how processes can accelerate or decelerate quickly over time. Learning how to discern and employ exponential functions equips students with an essential mathematical toolkit.

Algebra 2

Algebra 2 is an advanced high school mathematics course that delves into complex concepts including exponential functions, quadratic equations, polynomial expressions, and more. This course is often a pivotal stage in a student's journey, introducing more abstract algebraic techniques and applications.

Key focuses in Algebra 2 include:

• Deepening understanding of functions and function behavior, including exponential functions.
• Exploring new types of equations and their solutions, such as rational and radical equations.

By the time students complete Algebra 2, they should feel comfortable analyzing various forms of functions, recognizing patterns, and applying algebraic reasoning to real-world problems. This subject is essential for building a foundation for further studies in calculus and other advanced fields of mathematics. Its role is crucial in equipping students with the skills and confidence needed to tackle higher-level math challenges.

Base of Exponent

The base of an exponent in an exponential function is a fundamental component determining the behavior of the function. In an expression like \( y = a \cdot b^x \), the \( b \) is what we call the 'base'.

Here's why the base is important:

• If \( b > 1 \), the function models exponential growth, meaning values will increase as the exponent increases.
• If \( 0 < b < 1 \), the function illustrates exponential decay, meaning values diminish as the exponent increases.
• If \( b = 1 \), the function is constant as changes in \( x \) have no effect on \( y \).

Understanding and identifying the base allows students to quickly determine how the function behaves without the need to graph it. Thus, the base is not just a number; it's a critical factor that influences exponential function dynamics. Mastering the base's role enables learners to grasp the essence of exponential changes and apply this knowledge to academic and real-world scenarios.