a = ≒15.37

b = 609-1955×15.37=29439.35

Which gives the equation of y=15.37x−29439.35 as below:

Graph 2: Population of China 1950~1995 with linear function

It appears to be rather fit to the coordinates. However, the coordinates above show to contain curves, which linear function cannot do, as it did not go through all the points. Also, if it happens to be linear function, it would mean that China’s population would grow infinitely at a constant speed.

Therefore, I have the conclusion that linear function is not the model of the graph.

Quadratic Function: y=ax2 +bx+c

Quadratic function has three parameters, in which I will calculate them out below.

I use (1955, 609), (1985, 1070) and (1960, 657.5) to find the parameters. These sets are chosen because they are the multiples of 5 or even integers, which are easier to calculate.

609=(1955)2 a+1955b+c

657.5=(1960)2 a+1960b+c

1070=(1985)2 a+1985b+c

It would be difficult to calculate if I use the method with linear function. Therefore, I use matrix to find the parameters as below shown:

=

In which after I inverse the equation, the parameters will become the main subject:

= using Graphic Digital Calculator

a≒0.23

b≒-877.70

c≒850186.83

Which gives the equation of y=0.23x2-877.7x+850186.83, as below:

Graph 3: Population of China 1950~1995 with quadratic function

According to the graph, we can see that quadratic function did not fit the points at all. Below is a recalculation by plotting variable x into the equation:

0.23(1955)2-877.7(1955)+850186.83= 13349.08 → off by 12740.08

0.23(1960)2-877.7(1960)+850186.83= 13462.83 → off by 12895.33

0.23(1985)2-877.7(1985)+850186.83= 14024.08 → off by 12954.08

From the recalculation, we could see that this equation has an inconstant parameter c, which it varies the value of y when adding itself onto the number, thereby unable to provide an appropriate curve.

I then used Graphic Digital Calculator to revise the equation, below is what the values given by the calculator:

y=ax2+bx+c

a=0.355454545

b=-124.7309394

c=108601.1173

It shows that the values of what I calculated are far from the correct value; this may be another reason of the result of the incorrect graph.

Quadratic Function II: y=a(x-b)2+c

I then decided to try another type of quadratic function. However, this time I tried to find it by using constant controller in Autograph to change the values of parameters in order to fit with the points.

Below is the result after serious of trials and errors:

Graph 4: Population in China 1950~1995 with quadratic function II

The values of the parameters are:

a=0.0784

b=1874.37

c=93

Which give the equation of y=0.0784(x-1874.37)2+93

The equation above fits better than the equation y=0.23x2-877.7x+850186.83, however the line of linear function is a better best fit than this one since it misses more points than linear function, such as coordinates from year 1970 to 1990 have all failed to meet the line of the best fit as they tend to be more inclined, compare to other coordinates.

By using constant controller, I was able to visually understand how the changes of the parameters can determine the shape of the graph. Therefore, I have come to an conclusion that quadratic function is not the ideal model for the given data, not only because the given data is shown not to be a perfect curve, thus the quadratic cannot fit onto it completely, but also it means that the population of China will become infinitively large at a constant increase.

Exponential Function: y=abx

For exponential function, I will use (1955, 609) and (1985, 1070) to find the parameters of this function, and to see if it will fit with the graph better.

609=ab1955

1070=ab1985

Since they both share the same value of a, therefore by making them into one equation we will have:

609b-1955=1070b-1985

First, divide 1070 from both sides, then divide b-1985, we get:

By simplifying the equation, we get:

609= a()1955

= a(8.93)15 a = 6.82-14

Which we get the equation of y= 6.82 1.02x as below:

Graph 4: Population of China 1950~1995 with exponential function

This model has the closest best-fit compare to other models being produced by other equations; it has managed to fit into most of the points.

However, if it were the model of this graph, it would mean that the population of China is infinitive and grows at a constant rate, which shouldn’t happen because from the graph, we could see that the slope of the graph is starting to change in year 1995, which may indicate the decrease of the population.

A researcher suggests that the population, P at time t can be modeled by:

, where K, L and M are parameters.

This is a simple logistic function.

When t=0, P(0)= , which K is at its maximum because it is indirectly proportional to parameter t.

Therefore, according to the given data from the above table, we can use GDC calculator to find the parameters:

K≒1946.18

L≒4.34

M≒0.33

By substituting the values of K, L and M into the formula, we have:

Graph 5: Population of China in logistic function

Here are additional data on population trends in China from the 2008 World Economic Outlook, published by the International Monetary Fund (IMF):

In which we have the following graph containing all the points:

Graph 7: The final population of China graph

I use the function and find the parameters using Graphic Digital Calculator:

K≒1617.46

L≒1.35

M≒0.399

Which I then plot the equation into the graph: