Problem 74 - A Graphical Approach to College Algebra

Question

The table lists annual enrollment \(y\) in thousands for the Head Start program \(x\) years after 1980 . $$\begin{array}{|c|c|c|c|} \hline x & 0 & 10 & 32 \\ \hline y & 376 & 541 & 1128 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Enrollment increased from 376,000 in 1980 to 1128,000 in 2012 with notable growth in enrollment after 1990.

Step 1: Identify the Variables

We have two variables: \(x\), which represents the number of years after 1980, and \(y\), which represents the enrollment in thousands. We are given specific values of \(x\) and their corresponding \(y\) values.

Step 2: Interpreting the Table Data

From the table, we have three data points: when \(x = 0\), \(y = 376\); when \(x = 10\), \(y = 541\); and when \(x = 32\), \(y = 1128\). These give us the annual enrollment for the years 1980, 1990, and 2012, respectively.

Step 3: Analyzing Trends or Patterns

To understand the data better, observe the trend in enrollment from one year to another. From 1980 (\(x = 0\)) to 1990 (\(x = 10\)), the enrollment increased by \(541 - 376 = 165\) thousand. From 1990 (\(x = 10\)) to 2012 (\(x = 32\)), the enrollment increased by \(1128 - 541 = 587\) thousand.

Step 4: Provide the General Interpretation

The overall trend shows an increase in enrollment over the years. Between 1980 to 1990, there is a growth of 165 thousand, and from 1990 to 2012, a larger growth of 587 thousand. These changes can be interpreted to understand how participation in the Head Start program expanded over these years.

Key Concepts

Data Interpretation

Data interpretation is the process of making sense of figures, comparing numbers, and understanding what they convey about a particular situation. When examining the table provided, we interpret the data by identifying the connection between the number of years since 1980 and the corresponding enrollment numbers for the Head Start program.
This table lists pairs of values for the variables given. Here, data interpretation involves reading these values to determine what they illustrate about enrollment over time.

• The first data point (when x = 0) tells us that in 1980, there were 376,000 enrollments.
• The second point (x = 10) indicates that by 1990, enrollments had increased to 541,000.
• The third point (x = 32) shows a further rise to 1,128,000 enrollments by 2012.

By interpreting these figures, we can conclude that there has been an upward trend in enrollment within these years. It's crucial to not just see the numbers but to understand their implications regarding growth and changes in participation over time.

Trend Analysis

Trend analysis examines changes in data over a period. It's about spotting upward and downward movements to predict future patterns.
In the Head Start program data, we see an increasing trend in enrollments. This tells us more than just numerical growth; it helps understand the nature and scale of changes over time. To spot specific trends, consider:

• From 1980 to 1990, enrollments increased by 165,000, suggesting steady growth.
• Between 1990 and 2012, there was a sharp increase of 587,000.

The steep rise from 1990 to 2012 could suggest a shift in policy, increased birth rates, or broader adoption of the program. By recognizing such leaps, trend analysis also proposes the effectiveness and success of the program in attracting more participants.

Dependent and Independent Variables

Understanding dependent and independent variables is pivotal when analyzing data. These concepts define how one variable may change in response to another. In our given dataset:

• The independent variable is years after 1980, denoted by x. It is called independent as it doesn't depend on other variables given in this scenario.
• The dependent variable is enrollment in thousands, denoted by y. This variable is dependent because its value changes based on the independent variable, years after 1980.

The independent variable acts as a foundation, while the dependent variable reflects the data change. Here, as time progresses, the enrollment number evolves, highlighting their cause-effect relationship. Thus, grasping this relationship helps clearly understand data trends and predict future changes.