Problem 66 - Algebra 2

Question

Identify each function as exponential growth or decay, and find the growth or decay factor. \(y=0.1 \cdot 2^{x}\)

Step-by-Step Solution

Answer

The function \(y=0.1 \cdot 2^{x}\) represents exponential growth and the growth factor is 2.

Step 1: Identify the Parameters

From the given function \(y=0.1 \cdot 2^{x}\), we can identify that \(a = 0.1\) and \(b = 2\).

Step 2: Determine the Type of Function

Since \(b = 2\) which is greater than 1, the function represents exponential growth.

Step 3: Find the Growth Factor

The growth factor is the base \(b\), which is 2 in this case.

Key Concepts

Exponential Growth

Exponential growth occurs when the rate of increase of a quantity is proportional to the current amount. This means that as the quantity grows, it does so at a continually increasing rate. In mathematical terms, an exponential growth function is presented in the standard form:

• \(y = a \, \cdot \, b^{x}\)

where \(y\) is the final amount, \(a\) is the initial amount or coefficient, \(b\) is the growth factor, and \(x\) represents time or another independent variable.

Example Explanation: In the exercise, \(y=0.1 \, \cdot \, 2^{x}\), the base of the exponential function is \(b = 2\), indicating that the rate of growth doubles for each increment in \(x\). Since \(b > 1\), the function signifies exponential growth.

Whenever you see an exponential equation with a base \(b\) greater than 1, it symbolizes that the quantity is growing rapidly.

Growth Factor

The growth factor in an exponential function is crucial to understanding how quickly the function grows. It is the number by which you multiply the current amount to find the next one as the variable \(x\) increases by 1. In the exponential function \(y = a \, \cdot \, b^{x}\), the growth factor is represented by \(b\).

• If \(b > 1\), the function shows growth, and your growth factor indicates how much you multiply with each step.
• If \(0 < b < 1\), it would indicate decay instead of growth.

In our exercise's function, \(y=0.1 \, \cdot \, 2^{x}\), the growth factor is 2. This means that for each unit increase in \(x\), the quantity \(y\) is doubled.

Quick Tip: Always look at the base of the exponent to quickly determine the rate of change. This number directly tells you how fast or slow the quantity is increasing.

Algebraic Functions

Algebraic functions are a broad category of mathematical expressions that involve variables and constants. They can include operations such as addition, subtraction, multiplication, division, and the application of exponents. Exponential functions like those showing exponential growth are a specific type of algebraic function.

Algebraic vs. Exponential Functions: While all exponential functions are algebraic, not all algebraic functions are exponential. What distinguishes exponential functions is that the variable is in the exponent, such as \(b^{x}\). This is what gives rise to their unique growth or decay properties.

• Exponential functions have constant growth rates.
• General algebraic functions may not follow such a constant growth pattern and often involve polynomial expressions, like \(ax^n + bx^{n-1} + \ldots\).

In the given problem, recognizing the function form helps you determine the behavior of the function efficiently. This grasp of differentiating exponential functions from other algebraic types can immensely aid in solving various mathematical problems effectively.