Problem 52 - 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. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -4 & -2 & 0 & 2 & 4 \\ \hline y & -1 & -1 & -1 & -1 & -1 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The data are linear with a slope of 0.

Step 1: Examine the Data Points Visually

The given points are (-4, -1), (-2, -1), (0, -1), (2, -1), and (4, -1). Notice that for each value of x, the value of y is the same: -1. Plotting these on a graph would show a horizontal line.

Step 2: Check for Linearity

Since the y-values do not change regardless of the x-values, this represents a constant function. In mathematical terms, it forms a horizontal line on the graph, which is linear. Thus, the data is linear.

Step 3: Determine the Slope m

For a linear function, the slope is determined by the change in y over the change in x. Here, since there is no change in y (it remains -1), the slope is 0. Mathematically, \( m = \frac{{\Delta y}}{{\Delta x}} = \frac{{0}}{{\text{any x-value difference}}} = 0 \).

Key Concepts

Slope of a line

The slope of a line is a fundamental concept in understanding linear functions. It describes how steep the line is and the direction in which it moves. The slope is calculated as the "rise over run," or more technically, the change in the y-values divided by the change in the x-values, given by the formula: \[m = \frac{\Delta y}{\Delta x}\] where \( m \) represents the slope, \( \Delta y \) is the change in the y-values, and \( \Delta x \) is the change in the x-values.
A positive slope indicates that the line rises as it moves from left to right, a negative slope means it falls, and a slope of zero signifies a horizontal line.
In our exercise, since all y-values are constant at -1, there is no vertical change, making \( \Delta y = 0 \). Therefore, \( m = 0 \), identifying our line as horizontal.

Constant function

A constant function is a special type of linear function that has a graph of a horizontal line. This means that no matter what the x-value is, the y-value remains constant. In the equation form of a constant function, it is expressed as:\[y = c\] where \( c \) is a constant number.
For our provided data, the y-values do not change; they consistently equal -1 for every x-value. Thus, the function is represented by the equation \( y = -1 \).
A constant function is always linear because the graph is a straight line—specifically, a line with zero slope. This shows consistency and uniformity, which can be particularly useful in modeling situations where output remains steady regardless of input.

Linearity

Linearity is a property of a relationship that indicates whether the relationship can be graphically represented as a straight line. Linear data shows a constant rate of change; however, not all linear functions show a visible change.
In mathematics, a function is considered linear if it can be graphed as a straight line on the Cartesian plane. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In the exercise context, the data points form a straight, horizontal line indicating linearity even though the change in y-values is zero (a constant function). This signifies not just the predictability but also the stability of the data.
It is important to recognize linearity in different sets of data for analysis purposes; it helps in constructing models and predicting outcomes.