Question

During a recent period of time, the number \(S\) (in thousands) of students enrolled in public schools in a certain country can be modeled by \(S=1.64 x^3-102 x^2+1710 x+36,300\), where \(x\) is time (in years). Use a graphing calculator to graph the function for the interval \(0 \leq x \leq 41\). Then describe how the public school enrollment changes over this period of time.

Short answer

The graphing calculator will give us a graph representing the change in public school enrollment by the thousands over the period of 41 years. The analysis of this graph should reveal the trends in enrollment.

Step-by-step solution

Understand the Function

The given function \(S=1.64x^3-102x^2+1710x+36,300\) is a cubic function, where \(x\) represents time in years, and \(S\) represents the number of students enrolled in public schools in the thousands.

Plot the Function

Using a graphing calculator, input the function and set the range of \(x\) from 0 to 41. This will give the graph of the function over the given interval.

Analyze the Graph

Once the function is plotted, we can see how the values of \(S\) change as \(x\) increases from 0 to 41. The shape and trend of the function gives us information about how the enrollment in public schools changes over this period.

Key concepts

Public School Enrollment

Understanding public school enrollment is essential when considering educational resources allocation, funding, and policy development. The given cubic function, \(S=1.64x^3-102x^2+1710x+36,300\), serves as a mathematical model representing this enrollment over time, where \(S\) indicates the number of students (in thousands) and \(x\) is the time in years. To demystify the behavior of the enrollment, one can follow the graph's curvature.

During the initial years (\(x\) close to 0), we observe a starting value of 36,300 thousand students. The enrollment trend is deciphered by looking at the cubic, quadratic, and linear components of the function. The cubic term \(1.64x^3\) suggests that over a longer period, the growth rate itself changes. The quadratic term (-102\(x^2\)) can often create a parabolic effect, indicating that there may be instances of increase and decrease in the rate of change of enrollment. Meanwhile, the linear term (1710\(x\)) adds a consistent rate of change.

By analyzing the graph, it's possible to discover periods of exponential growth, steady increases, declines, or even fluctuations in enrollment. This information is instrumental for stakeholders to project future needs and adaptations required in the public education system.

Graphing Calculator Usage

A graphing calculator is a versatile tool for students and professionals alike, allowing for complex computations, function plotting, and data visualization. In the context of analyzing a cubic function for public school enrollment, the graphing calculator becomes indispensable.

To graph the given function, users input the equation and define the interval of \(x\) from 0 to 41. This reflects the span of years over which the enrollment data is considered. Calculators often enable zooming and panning features, offering users the ability to closely examine specific portions of the graph, such as trends or turning points. This examination through graphing is a powerful form of modeling with algebra, providing visual insights into various factors affecting public school enrollment over time.

Furthermore, advanced graphing calculators can manipulate algebraic expressions, calculate derivatives, and evaluate integrals, which can deepen the understanding of the rates of change within the enrollment model. For students learning this aspect of algebra, hands-on practice with a graphing calculator consolidates their grasp of abstract concepts and applications in real-world scenarios.

Modeling with Algebra

Modeling with algebra is a potent mathematical approach for representing real-world phenomena, like enrollment trends, climate change, population growth, or financial forecasting. In the case of our public school enrollment model, algebra helps us pinpoint and interpret patterns and changes over time.

Algebraic models are structured on equations involving variables and constants. The cubic function \(S=1.64x^3-102x^2+1710x+36,300\) exhibits an algebraic representation of enrollment numbers, enabling us to predict future outcomes based on historical data. Modeling with algebra involves a rhythm of constructing the model, interpreting results, and refining the equation to enhance accuracy.

As part of the algebra curriculum, students are often encouraged to use these skills for reasoning and problem-solving. Analyzing the function’s increasing or decreasing segments, and inflection points, gives students the capacity to draw crucial inferences about enrollment patterns and propose data-driven decisions. Thereby, modeling with algebra serves as a bridge connecting abstract mathematical theories to tangible societal issues.