Problem 47 - Algebra 1: Concepts and Skills

Question

Classify the model as exponential growth or exponential decay. Then identify the growth or decay factor and graph the model. $$ y=112\left(\frac{2}{3}\right)^{t} $$

Step-by-Step Solution

Answer

The given function is an exponential decay function with a decay factor of \(\frac{2}{3}\)

Step 1: Identify the type of exponential function

The given function is \(y=112(\frac{2}{3})^{t}\). This function has a base of \(\frac{2}{3}\), which is less than 1. Therefore, the function represents an exponential decay.

Step 2: Identify the decay factor

The decay factor in an exponential decay function is the base of the exponent. In this case, the base of the exponent is \(\frac{2}{3}\). So, the decay factor for this function is \(\frac{2}{3}\).

Step 3: Graph the function

To graph \(y=112(\frac{2}{3})^{t}\), we need to plot some points. We can do this by choosing some values for \(t\) and then calculating the corresponding \(y\)-values. Then, we can plot these points on the graph and sketch a smooth curve through them. Since it's an exponential decay function, the curve will go downwards from left to right.

Key Concepts

Exponential Growth

Exponential growth describes a process where the quantity increases at a rate proportional to its current value.
This means that as time progresses, the rate of increase becomes faster and faster. The mathematical model for exponential growth is typically expressed as:

• \( y = a(1 + r)^t \)

Here, \( a \) represents the initial amount, \( r \) is the growth rate (expressed as a decimal), and \( t \) is time.
The term \( 1 + r \) is known as the growth factor. If the growth factor is greater than 1, it signifies that the quantity is expanding.An essential aspect of exponential growth is that it leads to rapid increases over time. This is common in processes like population growth, investment returns, and certain types of biological processes. Recognizing exponential growth is crucial for predicting future outcomes and understanding the potential impact of ongoing trends.

Exponential Decay

Exponential decay occurs when a quantity decreases at a rate proportional to its current value.
The model for exponential decay is defined as:

• \( y = a(1 - r)^t \)

In this equation, \( a \) stands for the original amount, \( r \) represents the decay rate (expressed as a decimal), and \( t \) is the time variable.
The term \( 1 - r \) is known as the decay factor, always being less than 1.A common scenario for exponential decay is radioactive decay, where the number of unstable atoms reduces over time. This concept is also applicable in scenarios like depreciation of assets, cooling of warm substances, and decrease of medication concentration in the bloodstream. Understanding exponential decay helps in strategically planning for situations with a diminishing trend.

Graphing Exponential Functions

Graphing exponential functions involves plotting points calculated from the function's equation and connecting these points to form a curve.
For an exponential decay function like \( y = 112(\frac{2}{3})^t \), we first choose various \( t \) values and compute the corresponding \( y \) values.

• For \( t = 0 \), \(y = 112(\frac{2}{3})^0 = 112 \times 1 = 112\).
• For \( t = 1 \), \(y = 112(\frac{2}{3}) = 74.67\) (approximately).
• For \( t = 2 \), \(y = 112(\frac{2}{3})^2 = 49.78\) (approximately).

These points are then plotted on a graph with the \( t \)-axis as horizontal time and the \( y \)-axis as the vertical measurement.
Exponential decay graphs slope downwards from left to right, showing how the quantity decreases over time. This visualization helps to see how quickly the function approaches zero but never quite reaches it.