Problem 457 - Precalculus

Question

For the following exercises, refer to Table 4.27. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {555} & {383} & {307} & {210} & {158} & {122} \\\ \hline\end{array}$$ 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 \( f(x) = 671.24 \times 0.73^x \).

Step 1: Select the Regression Model

Since the problem asks for an exponential function, we use the model \( f(x) = a imes b^x \), where \( a \) and \( b \) are constants to be determined from the data.

Step 2: Linearize the Data

To apply linear regression, we take the natural logarithm of the function values. Set \( y = \log(f(x)) \). The transformed data is:\[\begin{array}{|c|c|} \hline x & y \hline 1 & \log(555) \ 2 & \log(383) \ 3 & \log(307) \ 4 & \log(210) \ 5 & \log(158) \ 6 & \log(122) \hline\end{array}\]

Step 3: Perform Linear Regression

Assume the linear form \( y = mx + c \), where \( m \) and \( c \) are the slope and the intercept, respectively. Calculate \( m \) and \( c \) using the least-squares method to fit the line to the \( x \) and \( y \) data.

Step 4: Determine Constants a and b

From the linear regression equation \( y = mx + c \), we have \( \log(a) = c \) and \( \log(b) = m \). Thus, \( a = e^c \) and \( b = e^m \). Compute \( a \) and \( b \) using the values of \( c \) and \( m \) from Step 3.

Step 5: Formulate the Exponential Function

With calculated \( a \) and \( b \), the exponential function becomes \( f(x) = a imes b^x \). Use these values to write the exponential equation that fits the original data.

Key Concepts

Understanding Exponential Functions

Exponential functions form the foundation of many natural processes and phenomena. An exponential function is typically expressed in the form \( f(x) = a \cdot b^x \), where \( a \) represents the initial value or coefficient, and \( b \) is the base of the exponential function. This base \( b \) determines the rate of growth or decay depending on whether its value is greater than or less than 1. For example:

• If \( b > 1 \), the function describes exponential growth, indicating an increase over time.
• If \( 0 < b < 1 \), it illustrates exponential decay, showing a decrease over time.

Understanding these parameters helps us to analyze and predict behaviors in a vast array of fields, from biology to finance.

What is Linear Regression?

Linear regression is a statistical method used for modeling the relationship between a dependent variable and one or more independent variables. The simplest form is a linear equation with one independent variable: \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. This technique helps in:

• Determining trends over time.
• Predicting future values based on past data.
• Understanding the strength and direction of relationships between variables.

In the context of exponential regression, linear regression is employed to simplify exponential relationships by transforming them into linear forms, making it easier to work with and interpret real-world data.

Exploring the Least-Squares Method

The least-squares method is a cornerstone in regression analysis. It is used to find the best-fitting curve to a given data set by minimizing the sum of the squares of the differences between the observed and predicted values. Here's how it works:

• The difference between each observed value and its corresponding predicted value is squared. This helps in penalizing larger errors more than smaller ones.
• All these squared differences are added together to form a total sum of squares.
• The regression line is adjusted so that this total sum is minimized, resulting in the most accurate model for the data.

This technique is crucial in finding the optimal values of the parameters in our exponential model by ensuring that the estimated relationship is as close to the actual data points as possible.

Using Logarithmic Transformation

Logarithmic transformation is a handy tool when dealing with exponential functions in regression analysis. It simplifies exponential growth or decay into a linear form, making it more manageable:

• By taking the natural logarithm of the dependent variable values, exponential equations such as \( f(x) = a \cdot b^x \) can be linearized to \( \,\log(f(x)) = \,\log(a) + x\,\log(b) \).
• This transformation allows us to utilize linear regression techniques, making calculation of parameters straightforward.
• Once the linear parameters are determined, inverse transformations (exponentiation) are used to return to the original scale and deduce the exponential model.

Using logarithmic transformations bridges the gap between exponential and linear relationships, facilitating easier data analysis and interpretation.