# In this Internal Assessment, functions that best model the population of China from 1950-1995 were thoroughly investigated.

MATHEMATICS SL

INTERNAL ASSESSMENT TYPE II

POPULATION TRENDS IN CHINA

In this Internal Assessment, functions that best model the population of China from 1950-1995 were thoroughly investigated.

The following table shows the population of China from 1950 to 1995:

To begin, instead of using 1950, 1955, 1960, 1965, 1970, 1975, 1980, 1985, 1990, and 1995 for the Year values, 0, 5, 10, 15, 20, 25, 30, 35, 40, and 45 will be used during the investigation to facilitate analysis. Through the use of a Graphic Display Calculator (GDC), the following data points were plotted on a graph.

For easier interpretation and scrutinizing of the data points, Graphical Analysis 3 was used to provide a more understandable and vivid graph for analysis.

The graph seems to show trends also exhibited by linear graphs, quadratic graphs, and cubic graphs:

Linear Fit:

From the graph, it is apparent that from 20, or 1970 to 45, or 1995, the graph increases in a linear manner in approximately the same rate of change (slope). Therefore, a linear model could provide an efficient fit for the graph of the data points. Now, assuming a linear fit for the plot of the data points, the function would be in the form y=Mx + B. The parameters, in this case, will be M and B, while the variable will be x. Parameter M represents the slope, or rate of change of the data points. Parameter B, on the other hand, represents the y-intercept, or the initial data point. Regarding constraints, the linear function for the population data would be limited to the first quadrant. A linear fit applied to the data points via Graphical Analysis 3 follows:

Quadratic Fit:

From the graph, the data points also seem to display their rise in a parabolic manner. Therefore, a quadratic function could also provide an efficient fit for the graph of the data points. Assuming a quadratic fit for the plot of the data points, the function would be in the form y=Ax2+Bx+C. In order to define parameters clearly, the function y=Ax2+Bx+C can be represented as

y=A(x-H)2+K. Parameter A represents the horizontal stretch of the graph. Parameter H represents the horizontal translation of the function. Parameter K represents the vertical translation of the function. The coordinate (-H, K) represents the vertex of the function. Regarding constraints, the quadratic function for the population data would be limited to the first quadrant. A quadratic fit applied to the data points via Graphical Analysis 3 follows: