Problem 49 - 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. $$ \begin{array}{|c|c|c|c|c|c|}\hline\hline x & 0 & 1 & 2 & 3 & 4 \\ \hline y & -1 & 3 & 7 & 11 & 15 \end{array} $$

Step-by-Step Solution

Answer

The data are linear with a slope of 4.

Step 1: Understanding Linear Data

Data are considered linear if they form a straight line when plotted. Mathematically, this means the rate of change between each pair of consecutive data points (the slope \( m \)) is constant.

Step 2: Calculate Slope between Points

To determine if the data are linear, calculate the slope between each consecutive pair of points using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Calculate the slope from (0, -1) to (1, 3), which is \( m = \frac{3 - (-1)}{1 - 0} = 4 \).

Step 3: Verify Consistent Slope

Check the slope for each consecutive pair: 1. From (0, -1) to (1, 3): \( m = 4 \).2. From (1, 3) to (2, 7): \( m = \frac{7 - 3}{2 - 1} = 4 \).3. From (2, 7) to (3, 11): \( m = \frac{11 - 7}{3 - 2} = 4 \).4. From (3, 11) to (4, 15): \( m = \frac{15 - 11}{4 - 3} = 4 \).

Step 4: Conclusion Based on Slope

Since the slope \( m = 4 \) is consistent across all intervals, the data points form a straight line, confirming that the data are linear. The slope of the line is \( m = 4 \).

Key Concepts

Slope Calculation

Slope is a fundamental concept in mathematics, especially when dealing with linear equations. It represents how steep a line is and is usually denoted by the letter \( m \). When determining if a set of data points is linear, calculating the slope is essential.

The formula for finding the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula basically measures the change in the \( y \)-values divided by the change in the \( x \)-values. It's like asking "for every step you move to the right on the graph, how many steps do you move up or down?"

For example, when calculating the slope from the point (0, -1) to the point (1, 3), the calculation would go as follows:

• \( y_2 - y_1 \) becomes \( 3 - (-1) = 4 \)
• \( x_2 - x_1 \) becomes \( 1 - 0 = 1 \)
• The slope \( m \) is then \( \frac{4}{1} = 4 \).

This process is repeated for each pair of consecutive points to check for a linear pattern.

Consistent Slope

When dealing with linear data, a consistent slope across each interval is key to confirming that the data is indeed linear. This means that no matter which pair of consecutive data points you choose, the calculation of the slope should yield the same result.

In the exercise given, the slope calculation was repeated for several pairs of points, as shown:

• From (0, -1) to (1, 3): \( m = 4 \).
• From (1, 3) to (2, 7): \( m = \frac{7 - 3}{2 - 1} = 4 \)
• From (2, 7) to (3, 11): \( m = \frac{11 - 7}{3 - 2} = 4 \)
• From (3, 11) to (4, 15): \( m = \frac{15 - 11}{4 - 3} = 4 \)

Each calculation results in a slope \( m = 4 \). This consistent result is a hallmark of a linear function. Consistency in the slope indicates that the data points form a straight line, which is precisely what defines linearity.

If there were any variation in these slope values, the data would not be linear.

Mathematical Modeling

Mathematical modeling involves representing real-world problems using mathematical forms. Linear models are one of the simplest and most useful types of mathematical models. They allow us to predict values and trends based on the linear relationship between variables.

A linear model is often expressed in the form of an equation, \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept (the point where the line crosses the y-axis).

In our exercise, once we established that the slope \( m \) was 4, we created a basic linear equation to describe the data:

• The equation would look like \( y = 4x - 1 \), where -1 is the y-intercept.
• This model suggests that for every increase of 1 in \( x \), the \( y \) value increases by 4.

Mathematical models are powerful because they allow us to make predictions. If we were to add another data point with \( x = 5 \), using our model, we could predict that the corresponding \( y \) value would be 19 (since \( 4 \times 5 - 1 = 19 \)). This predictive power is why mathematical modeling is a valuable tool in various fields such as science, engineering, and economics.