91Ó°ÊÓ

Nearly linear or exponential data: One of the two tables below shows data that are better approximated with a linear function, and the other shows data that are better approximated with an exponential function. Make plots to identify which is which, and then use the appropriate regression to find models for both. $$ \begin{aligned} &\begin{array}{|c|c|} \hline t & f(t) \\ \hline 1 & 3.62 \\ \hline 2 & 23.01 \\ \hline 3 & 44.26 \\ \hline 4 & 62.17 \\ \hline 5 & 83.25 \\ \hline \end{array}\\\ &\begin{array}{|c|c|} \hline t & g(t) \\ \hline 1 & 3.62 \\ \hline 2 & 5.63 \\ \hline 3 & 8.83 \\ \hline 4 & 13.62 \\ \hline 5 & 21.22 \\ \hline \end{array} \end{aligned} $$

Step by step solution

01

Plot the Data

Start by plotting both sets of data. For the table containing \(f(t)\), plot the points (1, 3.62), (2, 23.01), (3, 44.26), (4, 62.17), and (5, 83.25). For \(g(t)\), plot the points (1, 3.62), (2, 5.63), (3, 8.83), (4, 13.62), and (5, 21.22). Look for patterns: a straight-line pattern suggests linear data, while a curved, upward-sloping pattern suggests exponential growth.

02

Identify Linear Data

After plotting, examine the graph. If the points of a dataset lie approximately in a straight line, that dataset is better approximated with a linear function. In this case, the data of \(g(t)\) appear linear.

03

Fit a Linear Model for Linear Data

Apply linear regression to the data set \(g(t)\). The linear regression model is of the form \( g(t) = at + b \). By calculating or using a calculator/computer, find the values of \(a\) and \(b\). For the dataset \(g(t)\), \(g(t) \approx 4.41t - 0.86\).

04

Identify Exponential Data

Examine the plot again and notice the pattern of \(f(t)\); it follows a curved upward pattern, indicating exponential growth.

05

Fit an Exponential Model for Exponential Data

An exponential function is of the form \( f(t) = a \cdot b^t \). By using exponential regression tools, calculate the parameters \(a\) and \(b\). For \(f(t)\), we find \(f(t) \approx 2.4 \cdot 1.8^t\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Function

A linear function is a mathematical expression in which the output grows proportionally to the input. It means that you can represent such a function using the equation \( y = mx + b \), where:

• \( m \) is the slope (the change in \( y \) for a unit change in \( x \)).
• \( b \) is the y-intercept (the point where the line crosses the y-axis).

In the context of data modeling, when data points lie approximately on a straight line, they are well-represented by a linear function. By plotting the points and fitting a linear function using linear regression, we can determine the best-fit line, as done with the dataset \( g(t) \). Here, the equation was found to be \( g(t) = 4.41t - 0.86 \), suggesting a direct and proportional growth over time with an established beginning point.

Exponential Function

Exponential functions, represented by the equation \( y = a \, b^x \), describe situations where the rate of change of the dependent variable accelerates over time or space. Key elements include:

• \( a \): the initial amount when \( x = 0 \).
• \( b \): the growth factor. If \( b > 1 \), the function indicates growth; if \( b < 1 \), it indicates decay.

In practical terms, exponential functions model phenomena where change occurs rapidly, such as population growth or compound interest.
In this exercise, plotting the dataset \( f(t) \) showed an upward curved pattern, hinting at its exponential nature. By applying exponential regression, the relationship was approximated as \( f(t) = 2.4 \, 1.8^t \). This demonstrates how each subsequent point grows by a factor of approximately 1.8, underscoring the power of exponential functions in capturing dramatic growth.

Data Modeling

Data modeling is the process of creating a mathematical representation of a set of data. It allows for predictions and insights by finding relationships and trends within the data. When modeling data:

• Start by plotting the data points to visually assess their pattern.
• Determine whether the data set follows a linear pattern or an exponential one.
• Choose the appropriate model (either linear or exponential) to apply based on observed patterns.
• Use regression techniques to quantify the relationship with the best fit parameters for the chosen model.

This exercise required distinguishing between linear and exponential data through plotting and identification, leading to the selection of the appropriate regression models. Regression analysis then provided the means to extract meaningful relationships from the data, demonstrating how effective data modeling can be in uncovering these patterns and trends.