Problem 2 - College Algebra with Corequisite Support

Question

Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.

Step-by-Step Solution

Answer

Yes, by checking if the base \(b\) is > 1 (growth) or 0 < \(b\) < 1 (decay).

Step 1: Identify the General Form of Exponential Function

An exponential function generally has the form \( f(x) = a \cdot b^x \), where \(a\) is a constant and \(b\) is the base of the exponential function.

Step 2: Determine Conditions for Growth or Decay

In an exponential function \( f(x) = a \cdot b^x \), whether the function represents exponential growth or decay depends on the base \(b\). If \(b > 1\), the function exhibits exponential growth. If \(0 < b < 1\), the function exhibits exponential decay.

Step 3: Examine the Base of the Exponential Function

Look at the base \(b\) in the exponential function formula. If \(b\) is greater than 1, the exponential function is growing. If \(b\) is between 0 and 1, the exponential function is decaying.

Step 4: Draw Conclusion from the Formula Only

Since we determine whether it is growth or decay by checking the value of \(b\), yes, it is possible to tell whether the function grows or decays just by looking at the formula \( f(x) = a \cdot b^x \).

Key Concepts

Exponential Growth

Exponential growth describes a process where the quantity increases rapidly over time. Think of how bacteria multiply. If one bacterium doubles every hour, in just a few hours, the number will have increased dramatically. This represents exponential growth.

• The key feature is the base of the exponential function, denoted as \( b \) in the formula \( f(x) = a \cdot b^x \).
• If \( b > 1 \), then the quantity grows exponentially, meaning it increases more and more rapidly as time goes on.

Essentially, the bigger the base, the steeper and faster the graph rises. In real-world scenarios, many phenomena such as population growth, compound interest, and the spread of viruses follow this pattern. Make sure to recognize \( b > 1 \) as the main indicator of exponential growth.

Exponential Decay

Exponential decay is the opposite of growth. It happens when a quantity reduces rapidly at a rate proportional to its current value. A classic example would be radioactive decay. Over time, the amount of radioactive substance decreases in an exponential manner.

• In the exponential function formula, if \( 0 < b < 1 \), the function describes exponential decay.
• This means that as time passes, the quantity gradually decreases towards zero, but never quite reaching it.

When dealing with exponential decay, think of common examples like depreciation in value, cooling of a hot object, or a leaking tap gradually emptying a bucket. The value of \( b \) falling between 0 and 1 leads to this downward trend in the graph.

Base of Exponential Function

The base of an exponential function, represented by \( b \) in the equation \( f(x) = a \cdot b^x \), is vital in determining the behavior of the function. This number dictates whether the function will express growth or decay.

• If \( b > 1 \), the function grows exponentially.
• If \( 0 < b < 1 \), the function decays exponentially.

The base is a multiplier that scales the rate at which the quantity increases or decreases. It's not just an arbitrary number but the core element that influences the "direction" of the function—upward for growth and downward for decay. Understanding how the base works is crucial, as it allows you to predict and interpret real-world phenomena modeled by these functions with ease.