Problem 58 - College Algebra with Modeling and Visualization

Question

Analyzing Real Data For the given data set complete the following. (a) Make a line graph of the data. Let this graph represent a function \(f\) (b) Decide whether \(f\) is linear or nonlinear. Median incomes of full-time male workers $$ \begin{array}{|c|c|c|c|c|}\hline \hline \text { Year } & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Income } & \$ 9,184 & \$ 19,173 & \$ 28,979 & \$ 37,435 \end{array} $$

Step-by-Step Solution

Answer

The function \(f\) is nonlinear.

Step 1: Recognizing the Variables

Identify the variables from the data set. Here, the variables are 'Year' and 'Income'. Year (1970, 1980, 1990, 2000) is the independent variable, and the incomes ( $9,184, $19,173, $28,979, $37,435) are the dependent variables.

Step 2: Plotting the Line Graph

Create a line graph using the given data. On the x-axis, plot the years (1970, 1980, 1990, 2000), and on the y-axis, plot the corresponding income values (\(9,184, \)19,173, \(28,979, \)37,435). Connect these points with straight lines to visualize the graph of function \(f\).

Step 3: Observing the Graph

Examine the plotted line graph. Notice how the line representing the data points is shaped. Check whether the slope between each point is constant or varies.

Step 4: Determining Linearity

Calculate the differences in income for consecutive years to see if the function's change is constant. - Increase from 1970 to 1980: \(19,173 - \)9,184 = \(9,989- Increase from 1980 to 1990: \)28,979 - \(19,173 = \)9,806- Increase from 1990 to 2000: \(37,435 - \)28,979 = $8,456Since the increases are not the same, the function \(f\) is nonlinear.

Key Concepts

Linear Functions

Linear functions are an essential concept in mathematics, representing relationships where one variable changes at a constant rate relative to another. In a linear function, every time the independent variable increases by a certain amount, the dependent variable changes by a uniform amount. This is characterized by the equation of a line, which is typically written in the form: \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept where the line crosses the y-axis.
In the context of our median income example, a linear function would imply that income increases by the same amount every decade. But when we calculated the differences in income between decades (e.g., from 1970 to 1980, or 1990 to 2000), they were not consistent. Therefore, this dataset doesn't exhibit linear characteristics.
Key characteristics to identify linear functions include:

• Constant rate of change or slope
• Straight-line graph representation
• An equation of the form \( y = mx + b \)

Understanding these traits helps in analyzing whether data presented in a real-world problem can be modeled with a linear function.

Nonlinear Functions

Nonlinear functions encompass relationships where the rate of change is not constant. This means increases or decreases can be unpredictable and do not follow a straight path. Nonlinear graphs can take various shapes—exponential curves, quadratic parabolas, or others that deviate from a straight line.
In our exercise, the median income data increases in differing increments over the years. From 1970 to 1980, the increase was different than that from 1980 to 1990, indicating a variable rate of change. This means the function describing this relationship is nonlinear.
Nonlinear functions have diverse applications, and some of their common features include:

• Varying rate of change
• Curved line graph representation
• More complex equations than just \( y = mx + b \)

Recognizing nonlinear characteristics can greatly assist in understanding complex real-world data trends and behaviors.

Line Graph Plotting

Plotting a line graph is a fundamental skill, useful for visually representing data relationships. A line graph uses a series of data points connected by straight lines, which helps identify trends over a period.
To plot a graph for the median income data set, you place the years on the x-axis and the corresponding incomes on the y-axis. By connecting these plotted points, you visualize the data as a continuous function \( f \).
Here are the simple steps involved in line graph plotting:

• Select your axes: Usually, time or sequences go on the x-axis, while dependent data values on the y-axis.
• Plot your points: Each data pair is a point on the graph.
• Connect your dots: Draw lines between the points to illustrate trends.

Visual assessments of the line can reveal whether the function is linear or nonlinear. In this case, the inconsistent spacing between the line’s points suggests a nonlinear function. Mastering line graph plotting enhances your data interpretation and presentation capabilities effectively.