Problem 552 - Precalculus

Question

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline x & {0.15} & {0.25} & {0.5} & {0.75} & {1} & {1.5} & {2} & {2.25} & {2.75} & {3} & { 3.5} \\ \hline f(x) & {36.21} & {28.88} & {24.39} & {18.28} & {16.5} & {12.99} & {9.91} & {8.57} & {7.23} & {5.99} & {4.81} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The data is best modeled by an exponential equation.

Step 1: Input Data into Graphing Utility

Begin by opening your graphing utility application, such as Desmos or a graphing calculator. Enter the given data points into the utility. The data consists of two sets of values with 'x' values: 0.15, 0.25, 0.5, 0.75, 1, 1.5, 2, 2.25, 2.75, 3, 3.5, and 'f(x)' values: 36.21, 28.88, 24.39, 18.28, 16.5, 12.99, 9.91, 8.57, 7.23, 5.99, 4.81. Each pair of (x, f(x)) represents a point on your graph.

Step 2: Create a Scatter Plot

Using the graphing utility, plot the points as a scatter plot. Each point on the plot correlates to a pair from the data table. Observe how the points are distributed across the graph to determine the shape of the scatter diagram.

Step 3: Analyze Scatter Shape

Examine the pattern of the points in the scatter plot. An exponential model would appear as a curve that increases or decreases rapidly and then flattens. A logarithmic model would start quickly and then slow as it increases. A logistic model would have an S-shaped curve. Look for the visual shape closest to these descriptions.

Step 4: Choose Regression Model

Based on observation, if the scatter plot decreases rapidly and approaches a horizontal asymptote, then an exponential regression model may be appropriate, reflecting exponential decay. The data has a decreasing nature, which suggests an exponential model.

Step 5: Perform Exponential Regression

Use the graphing utility's regression feature to fit an exponential model to the data. Select exponential regression and let the utility calculate the best-fit line. Ensure the coefficient values are rounded to five decimal places as necessary. The utility will generate an equation of the form: \(y = a \, e^{bx}\).

Step 6: Write Down Exponential Equation

Once the regression is complete, write down the equation displayed by the graphing utility. If the resulting equation, for example, is \(y = 38.50000 \, e^{-0.51700x}\), round all coefficients to five decimal places, confirming the proper exponential form.

Key Concepts

scatter plot

A scatter plot is a type of graph used to display and observe the relationship between two sets of numerical data. Each data pair from the two sets is represented as a point on the graph. In this particular example, each point on the scatter plot would correspond to one pair from the dataset where the 'x' value is plotted on the horizontal axis and the 'f(x)' value is plotted on the vertical axis.

Scatter plots are incredibly helpful because they visually illustrate the distribution and any potential correlation between the data points. If the plotted points seem to form a pattern, this pattern provides clues about the type of relationship that exists between the variables.

• If points rise or fall sharply at first and then level off, this might suggest an exponential relationship.
• If points start slow and then accelerate, a logarithmic relationship might be indicated.
• If the points form an S-shape, a logistic relationship could be the key.

The beauty of scatter plots lies in their ability to offer a visual clue to what kind of mathematical model might best describe the given data, guiding us toward the next steps in data analysis.

graphing utility

A graphing utility is a tool that helps you create graphs and plots with ease. Modern graphing calculators or software applications like Desmos are wonderful examples of such utilities. They simplify the process of entering data, plotting points, and performing mathematical analyses through visual representations.

Here’s how you can use a graphing utility effectively for an exercise like this one:

• Start by inputting the data points into the software or calculator. Ensure that all 'x' values and corresponding 'f(x)' values are entered accurately.
• Use the functions of the graphing utility to create a scatter plot, displaying all data points simultaneously.
• Explore different types of regression models that the application might offer.

Graphing utilities not only help to visualize the data in scatter plots but also support advanced analysis, such as running regressions to find the best-fitting mathematical models. They are indispensable for data analysis, particularly in finding relationships and testing hypotheses. Moreover, they often allow you to display and study equations right on the screen, providing further insight into the data's behavior.

data modeling

Data modeling is the process of using mathematical models to represent real-world data. The goal of data modeling is to capture the underlying patterns or trends in data, which can be used for forecasting or understanding complex phenomena. It's an essential skill when working with empirical data, as it helps you interpret and predict behaviors accurately.

Exponential regression is a particular form of data modeling useful when data exhibit behaviors such as decay or rapid growth. In this context, data modeling would allow you to identify a pattern of exponential decay, where the decrease becomes slower over time, ultimately stabilizing at a certain level.

The process of data modeling can be broken down into steps:

• Create a scatter plot to visually examine the data and identify the potential pattern.
• Select the proper regression model using your visual assessment and understanding of the data.
• Fit the data to the chosen model, such as exponential, by using tools like a graphing utility to find the equation that best represents the data's pattern.

Successful data modeling means achieving a model equation that accurately reflects the scatter plot of the data, which is crucial for making predictions or drawing conclusions about the data's behavior. The precision of these models can often be improved by rounding off any coefficients to a reasonable number of decimal places, typically five in this scenario, ensuring accuracy without unnecessary complexity.