Problem 51 - College Algebra with Modeling and Visualization

Question

Decide whether the data are linear or nonlinear. If the data are linear, state the slope \(m\) of the line passing through the data points. $$ \begin{array}{|c|c|c|c|c|c|}\hline\hline x & -5 & -3 & 1 & 3 & 5 \\ \hline y & -5 & -2 & 1 & 4 & 7 \end{array} $$

Step-by-Step Solution

Answer

The data is nonlinear as the differences in slopes are inconsistent.

Step 1: Understand the Problem

We need to determine if the data represented in the table fits a linear pattern (a straight line). For linear data, the change in "y" relative to the change in "x" should be consistent for all data points. If it is consistent, we then calculate the slope \(m\) of this straight line.

Step 2: Calculate Differences in Y-values

Calculate the differences between consecutive \(y\) values. \(-2 - (-5) = 3\), \(1 - (-2) = 3\), \(4 - 1 = 3\), \(7 - 4 = 3\). The differences are consistently 3.

Step 3: Calculate Differences in X-values

Calculate the differences between consecutive \(x\) values. \(-3 - (-5) = 2\), \(1 - (-3) = 4\), \(3 - 1 = 2\), \(5 - 3 = 2\). The differences aren't consistent, with both 2s and a 4.

Step 4: Verify Consistent Quotients

Divide each change in \(y\) by the corresponding change in \(x\) to find the slope \(m\) between each set of points. \(\frac{3}{2}\), \(\frac{3}{4}\), \(\frac{3}{2}\), \(\frac{3}{2}\).

Step 5: Assess Linearity

To be linear, each quotient should be equal. Here, the quotient \(\frac{3}{4}\) differs from \(\frac{3}{2}\), indicating that the pattern isn't linear with respect to the slope of a single line.

Key Concepts

Slope Calculation

The slope calculation is fundamental for understanding if a dataset demonstrates linear behavior. In the context of a straight line, the slope, denoted as \(m\), represents how steep the line is. Mathematically, the slope is defined as the ratio of the change in the y-values to the change in the x-values between two data points. Formulaically, it's expressed as \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the difference in y and \( \Delta x \) is the difference in x.
To compute the slope:

• First, subtract the y-value of the first point from the second point to get \( \Delta y \).
• Then, subtract the x-value of the first point from the second point to get \( \Delta x \).
• Finally, divide \( \Delta y \) by \( \Delta x \) to obtain the slope.

Understanding the slope helps in determining whether the plotted data points form a straight line across a graph.

Data Analysis

Data analysis involves inspecting and modeling data with the goal of discovering useful information, reaching conclusions, and supporting decision-making. For linear data analysis, one key aspect is assessing the consistency of data changes. This means evaluating if the change in y-values over consistent changes in x-values leads to a uniform slope.
An effective way to analyze this dataset is by reviewing the differences:

• Calculate the differences between consecutive y-values to see if they remain constant.
• Compute the differences between x-values.
• Check if the slope, calculated for different pairs of points, remains the same.

By ensuring these measures, you can decide if the data fits a linear model or not. Consistent slopes across the range affirm a linear trend.

Linearity Assessment

Linearity assessment is the process of determining if a set of data points form a straight line when plotted on a graph. It's a crucial step in identifying linear relationships in datasets. The core of linearity assessment is checking if the slope between every set of consecutive points is identical.
In our example, the consistent slope for most intervals suggested a linear pattern. However, one mismatch in slope indicated otherwise. This inconsistency prevents the data from being called linear. A true linear dataset will show the exact same slope (ratio of \( \Delta y \) to \( \Delta x \)) across all data points.
If even one interval shows a different slope, the data cannot be considered linear. Thus, linearity assessment requires thorough cross-verification of all data point pairs to establish whether they follow a uniform slope.