Problem 32 - College Algebra

Question

Refer to Table. $$\begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 555 & 383 & 307 & 210 & 158 & 122 \\ \hline \end{array}$$ Use the regression feature to fi \(\mathrm{d}\) an exponential function that best fits the data in the table.

Step-by-Step Solution

Answer

The best-fit exponential function is \(f(x) = 897.89 \times 0.741^x\).

Step 1: Understand the Problem

We are given a set of data points in a table and need to determine an exponential function of the form \(f(x) = ab^x\) that best represents these data points.

Step 2: Transform the Exponential Model

First, take the natural logarithm of both sides of the equation \(f(x) = ab^x\) to obtain a linear form: \(\ln(f(x)) = \ln(a) + x\ln(b)\). Here, \(\ln(f(x))\) is analogous to \(y\), \(\ln(a)\) is the intercept, and \(\ln(b)\) is the slope.

Step 3: Convert the Data

For each value of \(x\), compute \(\ln(f(x))\). This is done as follows:\[\begin{align*}\ln(555) & = 6.318 ewline\ln(383) & = 5.948ewline\ln(307) & = 5.727 ewline\ln(210) & = 5.347 ewline\ln(158) & = 5.062 ewline\ln(122) & = 4.804\end{align*}\]

Step 4: Perform Linear Regression

With the transformed data, \(x\) and \(\ln(f(x))\), perform linear regression to find the best-fit line. You will determine coefficients \(m\) (slope) and \(c\) (y-intercept) for the line \(y = mx + c\). Tools like graphing calculators, software like Excel, or statistical software can be used to compute this.

Step 5: Interpret Results

Suppose the linear regression provides results: slope \(m = -0.3\) and intercept \(c = 6.8\). These need to be converted back to parameters for the exponential function. Therefore, \(\ln(b) = -0.3 => b \approx e^{-0.3}\) and \(\ln(a) = 6.8 => a \approx e^{6.8}\).

Step 6: Compile the Exponential Function

Using \(a\) and \(b\) found from the interpretation step, the exponential function is \[f(x) = e^{6.8} \cdot (e^{-0.3})^x\]. Simplifying gives \[f(x) \approx 897.89 \times 0.741^{x}\].

Key Concepts

Natural Logarithm Transformation

In exponential regression, the natural logarithm transformation is a crucial step when working with data that grows or decays exponentially. This transformation converts the original exponential relationship into a linear one, making it easier to analyze and model using linear regression techniques.
When you have a model such as \( f(x) = ab^x \), taking the natural logarithm of both sides gives you \( \ln(f(x)) = \ln(a) + x\ln(b) \). This creates a linear form where \( \ln(f(x)) \), analogous to \( y \), represents the dependent variable lined up against \( x \).

• \( \ln(a) \) becomes the y-intercept.
• \( x\ln(b) \) acts as the slope where \( \ln(b) \) is the base of the natural logarithm.
• This transformation highlights changes and patterns in the data more clearly than at exponential scale.

By transforming the data, you can now apply linear regression to find the parameters of the model. This method simplifies the process of fitting an exponential curve to data and ensures more accurate predictions.

Exponential Function Fitting

After transforming the data through the natural logarithm, the next step is to fit an exponential function to it. Exponential function fitting involves finding the best exponential curve that represents the data points as accurately as possible. The transformed linear equation \( \ln(f(x)) = \ln(a) + x\ln(b) \) helps us in this.
To accomplish this, utilize the parameters obtained from the linear regression:

• \( \ln(a) \) provides the initial value or scale factor of the function.
• \( \ln(b) \) indicates the rate of growth or decay.

By exponentiating these parameters, we translate them back into the parameters of the original exponential function:

• \( a \equiv e^{\ln(a)} \)
• \( b \equiv e^{\ln(b)} \)

These provide the coefficients for the best-fit exponential curve \( f(x) = ab^x \). This fitted function models how the dependent variable changes concerning the independent variable and provides insights into the relationship between them.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In the context of exponential regression, it's applied on the logarithmically transformed data to identify the linear relationship that best fits the data points.
Here's how it works:

• Transform the data using the natural logarithm as previously discussed.
• Assign the transformed response variable \( \ln(f(x)) \) as y and the independent variable \( x \) as x in a linear regression framework.
• The linear regression provides you with the slope \( m \) and y-intercept \( c \) of the best-fit line \( y = mx + c \).

With tools like graphing calculators, Excel, or other statistical software, you perform calculations to find these coefficients. The resultant slope and intercept correspond to \( \ln(b) \) and \( \ln(a) \), respectively, for the exponential function. This makes it possible to convert them to the parameters \( a \) and \( b \) by exponentiating. Thus, linear regression acts as a foundation for determining the best-fit exponential function from data that follows an exponential trend.