Problem 19 - College Algebra

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. $$ (0,2000) \text { and }(2,20) $$

Step-by-Step Solution

Answer

The equation represents exponential decay because the base \( b = 0.1 \) is between 0 and 1.

Step 1: Understand the problem

We are given two points on a curve, (0,2000) and (2,20), and need to determine if the curve represents exponential growth, decay, or neither.

Step 2: Identify the exponential function

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial quantity and \( b \) is the growth (\( b > 1 \)) or decay factor (\( 0 < b < 1 \)).

Step 3: Check initial value

Observe that the point (0,2000) indicates that \( a = 2000 \) when \( x = 0 \). This confirms that the function passes through this initial value.

Step 4: Use the second point to determine the type of exponential relationship

Substitute the point (2,20) into the function. That gives us 20 = 2000\(b^2\). To find \( b \), we isolate it: \( b^2 = \frac{20}{2000} = 0.01 \), so \( b = \sqrt{0.01} = 0.1 \).

Step 5: Classify the equation

From the previous step, we found \( b = 0.1 \), which is less than 1 but greater than 0. This implies exponential decay.

Key Concepts

Exponential Growth

Exponential growth happens when an initial quantity increases over time at a constant rate. This rate of increase is determined by the base, \( b \), in the exponential function equation, \( y = ab^x \). If \( b > 1 \), the function represents exponential growth.
Here's a quick breakdown of how it works:

• Starts with an **initial value**: \( a \) is the starting amount when \( x = 0 \).
• The **growth factor** \( b \): A value greater than 1, indicating that as \( x \) increases, \( y \) increases at a rate that multiplies the initial value.
• The function increases rapidly: Because each time \( x \) increases by one, \( y \) is multiplied by \( b \).

Real-life examples of exponential growth include the population of a species under ideal conditions or compound interest, where money increases over time. If in our exercise, the value of \( b \) was greater than 1, we'd observe an increase from the initial value of 2000.

Exponential Decay

Exponential decay is the opposite of growth; it describes a process where the quantity decreases over time. This is indicated by the base of the exponential function, \( b \), being between 0 and 1. From our exercise solution, we determined \( b = 0.1 \), a classic sign of decay.
Key characteristics of exponential decay:

• **Initial value** \( a \): This is still the starting amount when \( x = 0 \), known as the initial value. Here, it's 2000.
• **Decay factor** \( b \): A number between 0 and 1 shows that as \( x \) increases, \( y \) decreases, affecting the initial value at a diminishing rate.
• The function decreases rapidly: With each increase in \( x \), \( y \) is multiplied by a fraction, causing a rapid decrease over time.

Examples in real life include radioactive decay and depreciation of assets, such as vehicles. In our specific exercise, \( b = 0.1 \) means that with each increase in \( x \) by 1, \( y \) is only 10% of the previous value.

Initial Value in Exponential Functions

In exponential functions, the initial value, \( a \), plays a crucial role. It provides the starting point of the function when the input \( x \) is zero. This value sets the stage for either growth or decay based on the base \( b \). In our exercise, the initial value is given by the point \( (0,2000) \), so \( a = 2000 \).
Important features of the initial value:

• The initial point \( a \) sets where the function begins. It is the value \( y \) when \( x = 0 \).
• Acts like a multiplier: Whether the function represents growth or decay, the initial value sets the scale of the function in relation to actual quantities.
• Influences computations: Determines how large or small the function's values will be, especially important for real-life applications like calculating money, population sizes, or the concentration of substances over time.

Understanding the initial value helps you comprehend how exponential changes take place from a known point, aiding significantly in the interpretation of function behavior either towards growth or decay.