Problem 57 - A Graphical Approach to College Algebra

Question

The following table shows the revenue in billions of dollars for iTunes in various years. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2008 & 2009 & 2010 & 2011 & 2012 \\ \hline \begin{array}{l} \text { Revenue } \\ \text { (\$billions) } \end{array} & 6.0 & 7.0 & 9.5 & 11.8 & 15.7 \\\\\hline\end{array}$$ (a) Use exponential regression to approximate constants \(C\) and \(a\) so that \(f(x)=C a^{x-2008}\) models the data, where \(x\) is the year. (b) Support your answer by graphing \(f\) and the data.

Step-by-Step Solution

Answer

Exponential model: \(f(x) = 6.0 \times 1.277^{x}\) predicts the data by fitting an exponential growth around the given data points.

Step 1: Assign Data Points

Assuming 2008 as year 0, assign the data points: \((0, 6.0), (1, 7.0), (2, 9.5), (3, 11.8), (4, 15.7)\). Convert years into numerical values starting from 0 for easier regression computations.

Step 2: Exponential Regression Formula

Use the general form of an exponential function, \(f(x) = C a^x\), where \(x\) is the number of years from 2008. We need to find constants \(C\) and \(a\) such that the sum of squared residuals \(\sum ( y_i - C a^{x_i} )^2\) is minimized, where \((x_i, y_i)\) are the data points.

Step 3: Calculate Regression Parameters

Using statistical software or a calculator with exponential regression capability, input the data points to find the best-fit parameters \(C\) and \(a\). The outputs will be \(C \approx 6.0\) and \(a \approx 1.277\), because these values minimize the sum of the squared differences between actual data points and the points predicted by the model.

Step 4: Construct the Exponential Function

Based on the calculated regression parameters, the approximate exponential function is \(f(x) = 6.0 \times 1.277^{x}\), where \(x\) is the year number starting from 0 (2008).

Step 5: Graph the Function on a Coordinate Plane

Plot the data points \((0, 6.0), (1, 7.0), (2, 9.5), (3, 11.8), (4, 15.7)\) on a coordinate plane. Use the function \(f(x) = 6.0 \times 1.277^{x}\) to plot the graph representing the exponential growth. Make sure to label axes appropriately with \(x\) as years since 2008 and \(y\) as revenue in billion dollars.

Step 6: Verify Graph Matching

Observe if the plotted exponential curve closely follows the data points. The curve should progressively increase and fit through or near the data points, indicating the model's accuracy in representing the revenue growth trend.

Key Concepts

Data Modeling

Data modeling involves using mathematical models to represent real-world phenomena. When dealing with data that shows growth or decline at a consistent rate, exponential models often provide a good fit. In our example, iTunes revenue increases every year. By using exponential regression, we create a model that fits the revenue data across years:

1. **Define Your Data Points:**
Start with the collected data. In this problem, the years are converted to numerical values from zero (starting from 2008). This results in data points like (0, 6.0), (1, 7.0), and so forth.

2. **Choose the Right Model:**
Exponential models are useful for data that exhibits proportional growth. They are expressed in the form: \(f(x) = C a^x\).

3. **Apply Regression Techniques:**
Using software or tools, plug in your data points to determine constants \(C\) and \(a\). These constants shape the model, optimizing the fit to the data.

Through proper data modeling, the complicated trends from numbers are simplified into functions that are easy to manipulate and understand.

Exponential Functions

An exponential function is a mathematical expression where a variable is in the exponent. These functions are crucial for modeling situations with rapid growth or decay. Understanding their structure helps accurately represent phenomena like iTunes revenue growth.

In our scenario, the exponential function is represented as:
\(f(x) = C a^{x-2008}\).

1. **Components of the Function:**
- **Initial Value (C):** This is the starting point of the function, similar to the initial revenue. - **Growth Factor (a):** Indicates how much the quantity multiplies as x increases. A value greater than 1 suggests growth, while a value between 0 and 1 suggests decay.

2. **Why Exponential?**
They suit data with constant growth rates, reflecting patterns where amounts compound, like finance and population growth.

3. **Function Behavior:**
As x increases, the function will rise or fall exponentially, depending on whether \(a\) is greater than or less than one.

This mathematical construct helps economists, scientists, and analysts predict future trends based on current data.

Graphing

Graphing is an effective way to visualize exponential growth or decay in data. By plotting data points and overlaying the exponential function, one can easily observe the fit and accuracy of the model.

1. **Plotting the Data Points:**
Each point represents the revenue for a specific year. For this exercise, the points include (0, 6.0), (1, 7.0), etc.

2. **Drawing the Exponential Function:**
Using the exponential model function \(f(x) = 6.0 \times 1.277^x\), we plot the curve. It should smoothly connect near or through these data points.

3. **Analyzing the Graph:**
By comparing the function's curve with actual data, you evaluate how well the model captures real growth patterns. Ideally, the curve should progress closely with the plotted points.

This visual representation provides a clear and intuitive understanding of the exponential trends, aiding both interpretation and communication of data insights.