Problem 54 - 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. $$ \begin{array}{|c|c|c|c|c|c|}\hline\hline x & 10 & 20 & 25 & 35 & 40 \\ \hline y & 40 & 190 & 300 & 600 & 790 \end{array} $$

Step-by-Step Solution

Answer

The data is nonlinear.

Step 1: Identify the Data Points

Start by writing down all the points provided in the table. The points are (10, 40), (20, 190), (25, 300), (35, 600), and (40, 790). These are the xy-coordinates taken from the table.

Step 2: Check for Constant Slope

To determine if the data is linear, compute the slope between consecutive points and see if it's constant. The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Calculate the slope for each pair.

Step 3: Calculate Each Slope

- Between (10, 40) and (20, 190): \(m = \frac{190 - 40}{20 - 10} = \frac{150}{10} = 15\).- Between (20, 190) and (25, 300): \(m = \frac{300 - 190}{25 - 20} = \frac{110}{5} = 22\).- Between (25, 300) and (35, 600): \(m = \frac{600 - 300}{35 - 25} = \frac{300}{10} = 30\).- Between (35, 600) and (40, 790): \(m = \frac{790 - 600}{40 - 35} = \frac{190}{5} = 38\).

Step 4: Analyze Slope Consistency

The slopes calculated between consecutive points are 15, 22, 30, and 38, indicating the slope is not constant. If the data were linear, these values should all be the same.

Step 5: Conclusion on Linearity

Since the slopes between consecutive points are different, the data is not linear. Therefore, there is no single slope \(m\) that applies to all the data points.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is a fundamental algebraic concept. The slope of a line represents its steepness, or rate of change, and is symbolized by the letter \(m\). Calculating the slope involves finding the difference in the \(y\)-values divided by the difference in the \(x\)-values between two data points.

• Start with two points, \((x_1, y_1)\) and \((x_2, y_2)\), from your data.
• Use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula helps identify the rate at which \(y\) changes with \(x\). If you calculate the slope between each pair of consecutive points in a dataset, a consistent slope value indicates linearity. However, if these slope values vary, the data is nonlinear as was demonstrated while analyzing the given data points.

Data Analysis

In the analysis of data, distinguishing between linear and nonlinear trends can be a game-changer in understanding the relationships between variables. Linear data has a constant slope, meaning that for every unit increase in \(x\), \(y\) increases by the same unit each time.
For example, if every consecutive calculation of slope gives the same result, you can conclude that the trend is linear.
Analyzing the given data, each pair of coordinates produced slopes that were different from one another:

• Between (10, 40) and (20, 190), the slope was calculated as 15.
• Between (20, 190) and (25, 300), the slope was 22.
• Further comparisons delivered slopes of 30 and 38, respectively.

These differing values indicate a nonlinear relationship. Nonlinear data can be more challenging to work with because it doesn’t adhere to a simple line of best fit. This influences how predictions or calculations may be carried out in further analysis.

Algebraic Concepts

Grasping the algebraic foundation of concepts like linearity and slope prepares you for tackling more complex mathematical challenges. Many algebraic problems boil down to recognizing patterns and applying the correct formulae. With linear data, we're often dealing with straight-line equations, of the form \( y = mx + b \).
Here, \(m\) represents the slope, while \(b\) is the y-intercept, the point where the line crosses the y-axis.

• In real-world scenarios, linear equations can simplify the interpretation of data trends and projections.
• However, recognizing when data is nonlinear is crucial because it signals that the relationship between variables isn't straightforward.

The exercise demonstrated a key algebraic truth: without a constant \(m\), our data couldn't be represented accurately by this kind of linear equation. Understanding when these concepts do and do not apply is essential for effective data manipulation and analysis in algebraic contexts.