Problem 55 - 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. Toyota vehicles sold in the United States (millions) $$ \begin{array}{|r|c|c|c|c|}\hline \hline \text { Year } & 1998 & 2000 & 2002 & 2004 \\ \hline \text { Vehicles } & 1.4 & 1.6 & 1.8 & 2.0 \end{array} $$

Step-by-Step Solution

Answer

The function \( f \) is linear based on the graph.

Step 1: Plot the Data Points

Identify the data points from the table. The years are the x-coordinate, and the vehicles sold are the y-coordinate. The points you have are (1998, 1.4), (2000, 1.6), (2002, 1.8), and (2004, 2.0). Plot these points on a coordinate system with the year on the x-axis and the number of vehicles (in millions) on the y-axis.

Step 2: Draw the Line Graph

Connect the plotted data points with straight lines. You should see a set of line segments connecting each point from left to right. This visual representation provides a graphical form of the function \( f \).

Step 3: Analyze the Line Graph

Examine the graph you've drawn. Check whether the points form a straight line. If they do, the function is linear. If not, the function is nonlinear. In this case, the points (1998, 1.4), (2000, 1.6), (2002, 1.8), and (2004, 2.0) form a straight line.

Step 4: Conclude the Nature of the Function

Since the graph of the data points is a straight line, the function \( f \) is linear. A linear function means the rate of change between any two points is constant, which is observed here as the number of vehicles sold increases by 0.2 million consistently every two years.

Key Concepts

Linear Function

A linear function is a mathematical relation between two variables where the rate of change between them remains constant. Imagine if you were tracking how many Toyotas were sold in certain years; you’d expect the change from year to year to be consistent if the function is linear. A classic way to express this is through the equation of a line, often in the form of \( y = mx + b \), where:

• \( m \) is the slope or the "rate of change" that indicates how much \( y \) changes for a unit of change in \( x \).
• \( b \) is the y-intercept, which is where the line crosses the y-axis.

In the context of the exercise, we plotted data points stemming from sold vehicles at different years. The continuous increase by 0.2 million vehicles every two years shows a constant slope, proving that our function is indeed linear.

Analyzing Real Data

To analyze real data effectively, we need to approach it scientifically. This often involves collecting data, plotting it, and examining visual representations like line graphs. When you analyze data from real-world contexts, like vehicle sales over the years, it can be revealing. For our given dataset:

• We observe the trend and pattern of sales over time.
• The growing figures from year to year help us understand how growth is occurring in a linear manner.
• Analyzing such data can help in forecasting future patterns, assuming trends continue consistently.

Real data analysis helps us measure and understand phenomena accurately, forming the backbone of predictions and strategic decision-making.

Coordinate System

A coordinate system is essentially a grid used to determine the position of points. It consists of two perpendicular lines—usually called the x-axis (horizontal) and y-axis (vertical)—that meet at a point called the origin (0,0). This system is beneficial for translating data into a visual format that we can comprehend easily. In our exercise:

• The x-coordinate signifies time (the years when data was recorded).
• The y-coordinate represents the quantity of items sold (in this case, millions of vehicles).
• By plotting these coordinates, we can effectively visualize the data trend over time.

The coordinate system makes seeing trends such as linear relationships a lot more intuitive, as it gives a clear picture of how quantities change concerning each other.