Problem 65 - Algebra 2

Question

Identify each function as exponential growth or decay, and find the growth or decay factor. \(y=3 \cdot 4^{x}\)

Step-by-Step Solution

Answer

The function \( y = 3 \cdot 4^x \) is an exponential growth function with a growth factor of 4.

Step 1: Identifying the Function

The given function is \( y = 3 \cdot 4^x \). In this function, a = 3 and b = 4.

Step 2: Determine the Type of Function

As b (4) is greater than 1, we conclude that the function shows exponential growth.

Step 3: Finding the Growth or Decay Factor

The value of b is the growth or decay factor. So, in this case, the growth factor is 4.

Step 4: Final Identification and Conclusions

Therefore, the given function \( y = 3 \cdot 4^x \) is an exponential growth function with a growth factor of 4.

Key Concepts

Understanding Exponential Growth

Exponential growth refers to the process where the quantity increases by a consistent percentage over time. This means the growth accelerates as time moves forward. A common example of exponential growth in real life is population growth, where each generation is larger than the previous. In mathematical terms, an exponential growth function is represented as \(y = a \cdot b^x\), where:

• \(y\) is the value at time \(x\)
• \(a\) is the initial value
• \(b\) is the growth factor, with \(b > 1\)

Exponential growth happens when the base, \(b\), is greater than 1. Each time \(x\) increases by one unit, \(y\) is multiplied by \(b\). This rapid escalation differentiates exponential growth from linear growth.

Determining the Growth Factor

The growth factor in an exponential function indicates how much the quantity multiplies each step. It is the base \(b\) in the function \(y = a \cdot b^x\). When the growth factor is greater than 1, it confirms the function models exponential growth. Consider the function \(y = 3 \cdot 4^x\):

• Here, the growth factor \(b\) is 4.
• This means with each unit increase in \(x\), \(y\) becomes four times larger.

This characteristic allows exponential functions to model rapid changes effectively. Understanding the growth factor helps predict future values and understand the behavior of the function.

Function Identification in Exponential Equations

Function identification involves determining whether a function is growing, decaying, linear, or constant. For exponential functions, identification depends on the growth factor. In an equation like \(y = a \cdot b^x\):

• If \(b > 1\), it's exponential growth.
• If \(0 < b < 1\), it's exponential decay.

In the provided example, \(y = 3 \cdot 4^x\), the growth factor \(b\) is 4, which is greater than 1. Thus, it identifies as exponential growth. Identifying the function type helps in understanding the underlying process being modeled, whether it be growth, decay, or another trend.