Problem 482 - Precalculus

Question

Use a graphing utility to find an exponential regression formula \(f(x)\) and a logarithmic regression formula \(g(x)\) for the points \((1.5,1.5)\) and \((8.5,8.5) .\) Round all numbers to 6 decimal places. Graph the points and both formulas along with the line \(y=x\) on the same axis. Make a conjecture about the relationship of the regression formulas.

Step-by-Step Solution

Answer

Both regression formulas equate to the identity function \(y = x\) for the given points.

Step 1: Plot the Points

The given points are \((1.5, 1.5)\) and \((8.5, 8.5)\). First, plot these points on a graph. These points lie on the line \(y = x\).

Step 2: Find the Exponential Regression Formula

To find the exponential regression formula, use a graphing utility and enter the points \((1.5,1.5)\) and \((8.5,8.5)\). The tool will provide an equation of the form \(f(x) = a \cdot b^x\). Given these points lie on the line \(y = x\), the found equation will closely resemble \(f(x) = 1 \cdot e^{\ln(x)}\). Using the calculator, the result is \(f(x) = 1.5 \cdot \left(\frac{8.5}{1.5}\right)^{\frac{1}{7}} \)^x = 1x alternate the result \(f(x) = x\).

Step 3: Find the Logarithmic Regression Formula

For logarithmic regression, use the same graphing utility with the points. The utility will find an equation of the form \(g(x) = c + d\cdot \ln(x)\). By calculation, since these points lie on \(y=x\), it yields precisely \(g(x) = x\).

Step 4: Graph the Points and Formulas

On the same graph, plot the two given points, both regression models \(f(x)\) and \(g(x)\), and the line \(y = x\). As determined, both \(f(x)\) and \(g(x)\) are very close approximations of the line \(y = x\).

Step 5: Make a Conjecture

Since the points lie on the line \(y = x\), both exponential \(f(x)\) and logarithmic \(g(x)\) regression formulas match closely with \(y = x\). Hence, for this particular set of points, the exponential and logarithmic formulas are essentially equivalent to the identity function \(y = x\).

Key Concepts

Exponential Regression

Exponential regression is a statistical method used to model the relationship between a dependent variable and an independent variable where the data trends resemble an exponential curve. Essentially, this technique fits an equation of the form \( f(x) = a \cdot b^x \) to data points.In our exercise, we have two points: \((1.5, 1.5)\) and \((8.5, 8.5)\). Using a graphing utility, we seek an exponential equation that best fits these data points.
When the points perfectly lie on a line where \( y = x \), as they do here, the result of the exponential regression aligns closely with the line \( f(x) = x \). This is because any point \((a, a)\) on \( y = x \) satisfies an exponential expression of \( f(x) = a(e)^0 \).
In this specific scenario, the exponential regression confirms the underlying linear relationship.

Logarithmic Regression

Logarithmic regression is another form of curve fitting that is effectively used when the relationship between variables is best represented by a logarithm function, generally of the form \( g(x) = c + d \cdot \ln(x) \).For the points \((1.5, 1.5)\) and \((8.5, 8.5)\), when employing a graphing utility, we derive a logarithmic equation that attempts to track the inherent pattern in the points.
Given that both points lie directly on the identity line \( y = x \), the logarithmic regression equation also essentially becomes \( g(x) = x \).
This equivalence suggests that under specific conditions, such as points lying on \( y=x \), logarithmic regression yields results comparable to simple linear expressions, reinforcing the conceptual link between logarithmic functions and the linear function at the sample points.

Identity Function

The identity function, represented by \( y = x \), is a straightforward mathematical construct where every input equals its output. This function plots a straight line through the origin with a slope of 1. In the provided exercise, both given points

• \((1.5, 1.5)\)
• \((8.5, 8.5)\)

lie along this line. This observation is significant. Both the exponential and logarithmic regression analyses produce equations that match the identity function almost exactly.
It reinforces the notion that the line \( y = x \) can describe a variety of relationships between variables that follow a one-to-one correspondence.
This scenario highlights this foundational concept used extensively across mathematical and practical applications.

Graphing Utility

A graphing utility is an electronic tool, often a calculator or computer software, that performs calculations and plots graphs based on user input.
In our task, these tools are indispensable for running exponential and logarithmic regression analyses on the dataset consisting of the points \((1.5, 1.5)\) and \((8.5, 8.5)\). With a graphing utility, one can input points and immediately generate equations representative of their relationships, visually interpret them, and make mathematical conjectures. They provide key functions such as:

• Fitting curves to data
• Comparing multiple models
• Plotting complex equations

This not only speeds up the computational process but also enhances understanding by providing a visual interface which helps students see outcomes clearly.
As such, graphing utilities are crucial in both educational and professional environments.