Problem 23 - Algebra 2

Question

Determine whether each function represents exponential growth or decay. $$ y=2(4)^{x} $$

Step-by-Step Solution

Answer

The function represents exponential growth.

Step 1: Identify the Base of the Exponential Function

The given function is \( y = 2(4)^x \). In exponential functions of the form \( y = ab^x \), \( b \) is the base. Here, the base is \( 4 \).

Step 2: Analyze the Base

For an exponential function, if the base \( b > 1 \), the function represents exponential growth. If \( 0 < b < 1 \), it represents exponential decay. In this case, the base \( 4 \) is greater than 1.

Step 3: Conclusion on Growth or Decay

Since the base \( 4 > 1 \), the function \( y = 2(4)^x \) represents exponential growth.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions used to model a diverse range of real-world phenomena, such as population growth, radioactive decay, and compound interest. These functions take the general form of \( y = ab^x \), where \( a \) is the initial value (or coefficient), \( b \) is the base, and \( x \) is the exponent.
One key characteristic is how these functions rapidly increase or decrease compared to linear equations as \( x \) changes. The inclusion of the exponent \( x \) causes exponential functions to "grow" or "decay" much faster than simple addition or subtraction present in linear growth.
Applications of exponential functions can be seen in nature and sciences. For example, modeling the population size over time or predicting the spread of diseases. Recognizing the behavior of these functions is essential to making informed predictions and understanding the processes they represent.

Base of Exponential Functions

In the context of exponential functions, the base \( b \) plays a crucial role in determining whether the function represents growth or decay. It occupies the position \( b \) in the standard form \( y = ab^x \).

• If the base \( b \) is greater than 1, the function will depict exponential growth. This simply means that as \( x \) increases, \( y \) will increase at an accelerating rate.
• If the base \( b \) is between 0 and 1, the function shows exponential decay. Here, as \( x \) increases, \( y \) will decrease, but not at a constant rate. Instead, it will taper off more slowly as opposed to linear decay.

Choosing the correct base is vital for accurately depicting the behavior of the phenomenon under study. It informs about anticipated trends and helps in constructing precise models for predictions.

Exponential Decay

Exponential decay is a process where quantities decrease at a rate proportional to their current value. This occurs in exponential functions when the base \( b \) is between 0 and 1. As you increase the exponent \( x \), the overall value of \( y \) declines.
Such behavior is commonly observed in contexts like radioactive decay, depreciation of assets, or cooling of objects. For instance, in radioactive decay, the amount of substance decreases exponentially over time, and the half-life represents the time taken for the substance to reduce to half.
Unlike exponential growth, where quantities can grow without bound, exponential decay approaches zero, though never fully reaches it. It provides crucial insight into how quickly things like radioactive material or economic value diminish, which is invaluable in fields like science and finance.