Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, refer to \(\underline{\text { Table }} 7 .\) $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 1125 & 1495 & 2310 & 3294 & 4650 & 6361 \\ \hline \end{array} $$ Use the regression feature to find an exponential function that best fits the data in the table.

Step-by-Step Solution

Answer

The exponential function is obtained as \( f(x) = ab^x \), with values for \( a \) and \( b \) found using regression.

Step 1: Understand the Problem

We need to find an exponential function that best fits the given data. The general form of an exponential function is \( f(x) = ab^x \), where \(a\) and \(b\) are constants to be determined. Our goal is to use regression analysis to find these constants.

Step 2: Setup the Regression Model

We will apply a logarithmic transformation to linearize the exponential function. Start by taking the natural logarithm of both sides of the exponential equation: \( \ln(f(x)) = \ln(a) + x \ln(b) \). This transforms the model into a linear model with \( y = mx + c \), where \( y = \ln(f(x)) \), \( m = \ln(b) \), and \( c = \ln(a) \).

Step 3: Calculate the Transformed Variables

Convert the \(f(x)\) values to their natural logarithms: - \( \ln(1125) \approx 7.02 \)- \( \ln(1495) \approx 7.31 \)- \( \ln(2310) \approx 7.74 \)- \( \ln(3294) \approx 8.10 \)- \( \ln(4650) \approx 8.44 \)- \( \ln(6361) \approx 8.76 \).

Step 4: Perform Linear Regression

Now, use linear regression to find the best fit line \( y = mx + c \), with the transformed \( y \) values against \( x \). Calculate \( m \) (slope) and \( c \) (intercept) using least squares method.

Step 5: Solve for Exponential Function Parameters

Using the slope \( m \) and intercept \( c \) from the linear regression, exponentiate to return to the exponential model.- \( b = e^m \)- \( a = e^c \).

Step 6: Construct the Exponential Function

Using the values of \( a \) and \( b \), reconstruct the exponential function \( f(x) = ab^x \) that best fits the data.

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. The general form is given by \( f(x) = ab^x \), where \(a\) and \(b\) are constants, and \(x\) is the variable. This type of function is commonly used to model growth or decay processes, such as populations or investments.
In the context of modeling, the constant \(a\) represents the initial value when \(x = 0\), and \(b\) is the base of the exponent that dictates the rate of growth or decay:

• If \( b > 1 \), the function models growth.
• If \( 0 < b < 1 \), it models decay.

Understanding exponential functions is crucial for interpreting data that doesn't grow or shrink linearly but exponentially, meaning the change accelerates over time.

Regression Analysis

Regression analysis is a statistical method used to determine the relationship between a dependent variable and one or more independent variables. It helps in identifying trends and making forecasts based on data.

In the context of predicting an exponential function from data, regression analysis involves adjusting the parameters \(a\) and \(b\) to best fit the observed values. We utilize this technique to find a model that minimizes discrepancies between the model's predictions and actual data observations:

• This is achieved by optimizing the fit to minimize the sum of squared differences between predicted and observed values.
• It is essential for creating predictive models in various fields such as economics, biology, and engineering.

By fitting an exponential model, we can make informed predictions about future behavior based on past data.

Logarithmic Transformation

Logarithmic transformation is a technique used to transform an exponential relationship into a linear one, making it easier to analyze and model using linear regression techniques.
When applied to an exponential function \(f(x) = ab^x\), taking the natural logarithm of both sides results in a linear equation: \(\ln(f(x)) = \ln(a) + x\ln(b)\). This transforms the data into a format that can be easily fitted using linear regression:

• The logarithm function is applied to the dependent variable \(f(x)\), which helps in stabilizing the variance and making the distribution more normal.
• This transformation allows for straightforward calculation of the slope \(m = \ln(b)\) and intercept \(c = \ln(a)\).

After performing this transformation, the exponential relationship feels like a linear one, facilitating analysis and interpretation.

Linear Regression

Linear regression is a method for modeling the relationship between a scalar dependent variable and one or more independent variables linearly. In our task, it's applied to logarithmically transformed data:
Given the transformed function \(\ln(f(x)) = \ln(a) + x\ln(b)\), we perform linear regression to find the best values of \(m\) and \(c\) from the equation \(y = mx + c\):

• Here, \(y\) denotes the transformed dependent variable \(\ln(f(x))\).
• The goal is to determine the slope \(m\) and y-intercept \(c\) that minimally deviate from the given data points, using a method called least squares.

This process provides a line of best fit, enabling us to deduce \(b = e^m\) and \(a = e^c\). Linear regression allows us to harness a straightforward mathematical approach to solve complex relationships, which we use to revert back to our original exponential function with practical constants.