Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, find the formula for an exponential function that passes through the two points given. (0,2000) and (2,20)

Step-by-Step Solution

Answer

The exponential function is \( y = 2000 \cdot 0.1^x \).

Step 1: Understanding the Exponential Function

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value and \( b \) is the base of the exponential function. Our goal is to find \( a \) and \( b \) such that the function passes through the given points.

Step 2: Using the Point (0, 2000)

Plug the first point (0, 2000) into the function. When \( x = 0 \), \( y = a \times b^0 = a \). So, \( a = 2000 \). Thus, our exponential function starts as \( y = 2000b^x \).

Step 3: Using the Point (2, 20)

Now, plug the second point (2, 20) into the function. This gives us \( 20 = 2000b^2 \). Simplifying, we have \( b^2 = \frac{20}{2000} = 0.01 \).

Step 4: Solving for b

Take the square root of both sides to find \( b \). This gives \( b = \sqrt{0.01} = 0.1 \).

Step 5: Writing the Final Function

Substitute \( a = 2000 \) and \( b = 0.1 \) back into the general form. Thus, the exponential function is \( y = 2000 \cdot 0.1^x \).

Key Concepts

Exponential Growth and Decay

Exponential functions are powerful tools used to model real-world phenomena where growth or decay occurs at a constant percentage rate over time. These functions can represent growth, like population increase, or decay, like radioactive substances.

When we talk about exponential growth and decay, we are referring to how these functions behave over time.

• **Exponential Growth**: This occurs when the base, \(b\), of the exponential function is greater than 1. In this scenario, the function increases rapidly as \(x\) increases. A real-life example could be a bank account balance increasing due to compound interest.
• **Exponential Decay**: This happens when the base \(b\) is between 0 and 1. Here, the function decreases rapidly. An example is the reduction of a medication's concentration in the bloodstream over time.

The pivotal characteristic of exponential growth and decay is their consistent rate of change. The percentage change per unit time remains constant, which distinguishes exponential patterns from linear ones. Understanding the signs of \(b\) aids in predicting the behavior of the function.

Solving Exponential Equations

To solve exponential equations like \(20 = 2000b^2\), you'll need to isolate the variable representing the base, often through algebraic manipulation or by taking logarithms.

In this specific problem, to solve for \(b\), follow these steps:

• Start by isolating the term with \(b\): \(b^2 = \frac{20}{2000}\).
• Simplify this expression to find \(b^2 = 0.01\).
• To find \(b\), take the square root of both sides: \(b = \sqrt{0.01} = 0.1\).

An exponential equation's solution often involves making both sides of the equation comparable, either by breaking down the powers or using properties of logarithms.

These strategies help you effectively find the unknown base or other components. It's essential to check your work by plugging back into the original equation to verify accuracy.

Graphing Exponential Functions

When graphing exponential functions, such as \(y = 2000 \cdot 0.1^x\), you'll notice specific inherent characteristics.

Exponential function graphs show either upward growth or downward decay, and their shape heavily depends on the base \(b\). Here's what to look for:

• A function like \(y = a \cdot b^x\) will have a horizontal asymptote. For most functions, this is the line \(y = 0\). This means the curve approaches but never crosses this line.
• If \(b\) is less than 1, as in our equation, the graph will show exponential decay, sloping downwards to the right.
• The initial value \(a\) determines where the graph starts when \(x = 0\).

When graphing, use a few key points calculated from the equation to plot. In the problem at hand, with an initial point at (0,2000), you can plot additional points by substituting different values of \(x\) to see the exponential function's decline. This visual representation helps in understanding how quickly or slowly change is occurring at every point.