Problem 27 - College Algebra

Question

Refer to Table 7. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array}$$ Use the regression feature to fi \(\mathrm{d}\) an exponential function that best fits the data in the table.

Step-by-Step Solution

Answer

Find \( f(x) = ab^x \) using linear regression on transformed \( \ln(f(x)) \) against \( x \).

Step 1: Understanding Exponential Functions

An exponential function can be expressed in the form \( y = ab^x \) where \( a \) is the y-intercept and \( b \) is the base of the exponential function. Our task is to find \( a \) and \( b \) that best fit the given dataset.

Step 2: Linearizing the Data

To transform this problem into a linear form, take the natural logarithm of \( f(x) \) in the dataset. Let \( y = \ln(f(x)) \). This results in a linear equation \( y = \ln(a) + x \cdot \ln(b) \), where \( \ln(a) \) is the intercept and \( \ln(b) \) is the slope of the line.

Step 3: Calculate Natural Logarithm of Each \( f(x) \)

Calculate \( y = \ln(f(x)) \) for each data point. We have: \[\begin{align*} y_1 &= \ln(1125), \ y_2 &= \ln(1495), \ y_3 &= \ln(2310), \ y_4 &= \ln(3294), \ y_5 &= \ln(4650), \ y_6 &= \ln(6361). \end{align*}\]

Step 4: Perform Linear Regression on Transformed Data

Use the \( x \) values (1 through 6) and the \( y \) values (calculated from \( \ln(f(x)) \)) to perform linear regression and find the best-fit line. This will give you the coefficients \( \ln(a) \) and \( \ln(b) \).

Step 5: Calculate \( a \) and \( b \) from Linear Regression Result

From the linear regression, extract \( \ln(a) \) and \( \ln(b) \). Compute \( a = e^{\ln(a)} \) and \( b = e^{\ln(b)} \) using the exponential function to revert from the logarithmic transformation.

Step 6: Construct the Exponential Function

With \( a \) and \( b \) determined, write the final exponential function as \( f(x) = ab^x \). This is the function that best fits the original data based on the exponential model.

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to examine the relationship between variables. Here, we specifically look into an exponential relationship between variables. The technique helps in predicting the value of a dependent variable based on the value of at least one independent variable. In the context of the provided exercise:

• The dependent variable is the function values \(f(x)\).
• The independent variable is \(x\).

Regression analysis involves several steps which include transforming data, choosing a suitable model, and calculating parameters that best fit the model.
One of the objectives of regression analysis is to minimize the difference between observed values and the predicted values from the model. This difference is known as the residual and can be visualized as the distance between points on a graph and the regression line. Finding the optimal line that best minimizes these residuals helps derive a more accurate model.

Linear Regression

Linear regression is a fundamental method in statistics used to model the relationship between a dependent variable and one or more independent variables. In this exercise, to fit an exponential model, the problem is transformed into a linear one by taking the natural logarithm.
The linear form of an exponential function allows you to apply linear regression by considering the transformed model:

• Form: \( y = \ln(a) + x \cdot \ln(b)\)
• Intercept: \( \ln(a) \)
• Slope: \( \ln(b) \)

By plotting the \(x\) values against the \( \ln(f(x)) \) values, the regression line can be identified. Linear regression computes the best-fit line through these points by minimizing the sum of squared vertical distances of each point from the line. This approach gives the values of the coefficients, enabling calculation of the original exponential parameters.

Natural Logarithm

A natural logarithm is a specific logarithmic function which has the base \(e\) (approximately 2.71828). It is a vital tool in converting an exponential function into a linear form, making it easier to handle mathematically.
In the given exercise, we convert \(f(x)\) into \(y = \ln(f(x))\) to facilitate the use of linear regression.
The primary benefits of employing natural logarithms in this context include:

• Simplifying the model: transforms multiplicative relationships into additive relationships.
• Linearization: makes the computation easier by converting the exponential problem to a linear one.
• Ease of interpretation: helps in understanding the rate of change, especially in growth-related scenarios.

The natural logarithm reduces the complexity of the exponential function, making it a linchpin in the analysis and interpretation of data that follows exponential patterns.