Problem 452 - Precalculus

Question

For the following exercises, refer to Table 4.26. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {1125} & {1495} & {2310} & {3294} & {4650} & {6361} \\\ \hline\end{array}$$ Use the regression feature to find an exponential function that best fits the data in the table.

Step-by-Step Solution

Answer

The best fit exponential function is \( f(x) = 762.8 \times 1.5^x \).

Step 1: Understand the Problem

We need to find an exponential function of the form \( f(x) = a imes b^x \) that best fits the given data in the table.

Step 2: Set the Form of Exponential Function

The form of the exponential function is given by \( f(x) = a imes b^x \). We need to find the values of \( a \) and \( b \) using regression.

Step 3: Use Logarithms to Linearize the Data

Take the natural logarithm on both sides of the exponential equation to linearize it: \( ext{ln}(f(x)) = ext{ln}(a) + x imes ext{ln}(b) \). This equation is in the form \( y = mx + c \), which allows us to use linear regression techniques.

Step 4: Transform the Data

Convert the given function values \( f(x) \) into logarithmic form using the natural log: Calculate \( ext{ln}(f(x)) \) for each \( x \) value in the table.

Step 5: Linear Regression on Transformed Data

Perform linear regression on the transformed data set, treating \( x \) as the independent variable and \( ext{ln}(f(x)) \) as the dependent variable. Find the slope \( m \) and the intercept \( c \).

Step 6: Determine Parameters

From the linear regression output, \( m = ext{ln}(b) \) and \( c = ext{ln}(a) \). Calculate \( b = e^m \) and \( a = e^c \) to find the parameters for the exponential function.

Step 7: Write the Exponential Function

Substitute the values of \( a \) and \( b \) back into the exponential form \( f(x) = a imes b^x \) to find the function that best fits the data.

Key Concepts

Linear Regression

Linear regression is a fundamental statistical method used to model the relationship between two variables by fitting a linear equation to the observed data. This involves finding the best-fitting straight line through a set of points in a two-dimensional space. The equation of a line is generally given by \( y = mx + c \), where:

• \( y \) is the dependent variable, often what we're trying to predict.
• \( x \) is the independent variable, or the input.
• \( m \) is the slope of the line, representing the change in \( y \) for a unit change in \( x \).
• \( c \) is the y-intercept, the point at which the line crosses the y-axis.

To perform linear regression, we calculate the slope and y-intercept that minimize the distance between the line and all of the points, typically using the method of least squares. This process allows us to predict future data points based on the existing data.

Logarithmic Transformation

A logarithmic transformation is a useful technique for linearizing data that follows an exponential pattern. When we have an exponential function given by \( f(x) = a \times b^x \), we aim to transform it into a linear form that can be handled with linear regression.We achieve this by taking the natural logarithm of both sides of the equation, leading to:\[ \ln(f(x)) = \ln(a) + x \times \ln(b) \]This is similar to the linear equation \( y = mx + c \), where:

• \( \ln(f(x)) \) is analogous to \( y \).
• \( x \) remains \( x \).
• \( \ln(b) \) is similar to the slope \( m \).
• \( \ln(a) \) acts as the intercept \( c \).

By using logarithmic transformation, we can convert a non-linear relationship into a linear one, making it more manageable for analysis and modeling techniques like linear regression.

Exponential Function

An exponential function is a type of mathematical function frequently encountered in growth and decay processes. It is typically expressed in the form \( f(x) = a \times b^x \), where:

• \( a \) is a constant representing the initial value or vertical shift of the graph.
• \( b \) is the base of the exponential, determining the rate of growth or decay.
• \( x \) is the exponent, usually a variable representing time or another changing quantity.

Exponential functions have distinct characteristics, such as:

• They increase rapidly for values of \( b > 1 \) and decrease for \( 0 < b < 1 \).
• The graph of an exponential function is a curve that gets steeper or more shallow over the length of the x-axis.
• Common real-life applications include population growth, radioactive decay, and compound interest.

Understanding exponential functions is crucial for mathematically modeling processes that change at rates proportional to their current value.

Data Fitting

Data fitting involves finding a mathematical function that best represents the relationship between a set of given data points. When dealing with observed data, the challenge is often to identify a model that accurately captures the underlying pattern without overfitting or underfitting the data.In exponential regression, our goal is to fit an exponential function to our data. This requires a few key steps:

• Identifying the form of the model, such as \( f(x) = a \times b^x \).
• Linearizing the data if necessary, often through logarithmic transformation.
• Performing linear regression on the transformed data to find the best-fit parameters \( a \) and \( b \).
• Calculating the resulting exponential model using these parameters.

The quality of the fit is usually assessed by evaluating the residuals—the differences between observed values and those predicted by the model. A successful data fitting process provides a function that can be used to make predictions or understand trends within the data set.