Problem 38 - Algebra 2

Question

Writing Like a debt, a deficit is a negative amount of money. Explain how you would model a deficit that is growing exponentially. In \(y=a b^{c x},\) would the values of \(a\) and \(c\) be positive or negative? Would the value of \(b\) be greater than 1 or less than 1\(?\)

Step-by-Step Solution

Answer

In the case of a growing deficit, \(a\) will always be negative as it represents a negative amount of money (the initial deficit). Parameter \(b\) will be greater than 1, as the deficit is growing over time. Lastly, \(c\) will be positive, as with the passage of time, the deficit (negative value) grows increasingly negative.

Step 1: Identify the Starting Value (a)

In an exponential function, the parameter \(a\) is the initial value of the function. For a growing debt, this will be the starting amount of the deficit. Since a deficit is a negative amount of money, the value of \(a\) will always be negative.

Step 2: Identify the Growth Rate (b)

The parameter \(b\) in the exponential equation represents the growth rate of the function. If the deficit is growing (i.e., the amount of money owed is increasing), then the growth rate must be greater than 1. In other words, for each unit increase in \(x\) (e.g., each month or year), the debt increases by a factor of \(b\). So, the value of \(b\) is greater than 1.

Step 3: Identify the Time Coefficient (c)

The parameter \(c\) in the exponential function represents the coefficient of the time variable \(x\). In this scenario, as a debt is growing, the coefficient \(c\) should be positive. This is because as time progresses, the debt (which is represented as a negative value) becomes more and more negative, hence growing negatively.

Key Concepts

Deficit Modeling

When it comes to understanding deficit modeling in the context of exponential growth, think of a deficit much like a debt you owe. A deficit represents a shortage, a negative amount of money. In mathematical modeling, we represent transformations, like a growing deficit, using exponential functions.
To model a deficit that increases over time, we use an equation similar to: \( y = a \cdot b^{c \cdot x} \)

• \(a\) is the starting value, which is negative since it is a deficit.
• \(b\) is the growth factor, which will be more than 1.
• \(c\) is the time coefficient indicating the impact of time on the growth.

In simple terms, deficit modeling uses negative starting values to show the initial shortage and a positive growth factor to represent the increase of that shortage over time.

Growth Rate

The concept of growth rate is essential in exponential functions, such as the one used to model deficits. In our exponential equation \( y = a \cdot b^{c \cdot x} \), the parameter \(b\) reflects the growth rate of the deficit.
The critical point to remember with growth rate is:

• If \(b\) is greater than 1, it signifies an increase. For each passing unit of time, the deficit grows by a specific multiplying factor.
• In the context of a growing deficit, \(b > 1\) means that the amount of the shortage is expanding with time.

Thus, a growth rate greater than 1 implies that the debt is increasing, portraying how deficit amounts become larger as time flows.

Time Coefficient

The time coefficient, denoted as \(c\) in the exponential equation, plays a crucial role in how swiftly or slowly a deficit modifies through time. It determines the pace at which the deficit grows.
In the provided exponential function model \( y = a \cdot b^{c \cdot x} \), the coefficient \(c\) affects the time variable \(x\):

• A positive \(c\) value results in the deficit getting larger as more time passes, meaning the negative deficit value either becomes more negative or less negative slower overtime, based on the context.
• It essentially modulates how aggressively the shortage expands over each time unit.

Therefore, the time coefficient is pivotal in pinpointing the rhythm of increase within your deficit model, specifically how closely the deficit value adheres to growing exponentially as time leads on.