This investigation aims to look at the different functions that best model the population of China from 1950 to 1955. The following table shows the data given; this data was then graphed in order to give me a better idea of the data and its shape. The variables in this investigation are (P) for population in millions and (t) for the year that the data was from. Year| Population in millions| 1950| 554.8|
The graphs slope is increasing, therefore functions like linear, quadratic, cubic, logarithmic, exponential and piecewise are all viable options along with a few other functions with the right changes to the slope. All these functions would require changes to vertical stretch, horizontal stretch as well as vertical and horizontal translation in order for the basic equation to match up with the data. The best fits would be the cubic or the exponential because of the shape of growth; I choose to investigate the cubic function in relation to this data because of the curve in the slope. This curve is very similar to the curve in the basic cubic function with the right changes to the vertical stretch and translation as well as horizontal stretch the cubic function should match up to the original data well. Year| Population in millions| 0| 554.8|
I changed the values in the x because it greatly simplifies my numbers for the next step that I chose. I chose to analytically find the cubic function using an augmented matrix method. I chose the data points (5,609), (15,729.2), (25, 927.8) and (35, 1070) so that my points used would be spread out throughout the original data, not all of it would be from one small section of the original graph.
A X B A-1B= X
5352153152253252353352 51151251351 × abcd = 609.0729.2927.81070.0X=abcd -0.0224666671.403-8.738333333620.425
The basic form of a cubic equation is y=ax3+bx2+cx+d, taking the values found using the augmented matrix method the function P(t)=-0.022466667x3+1.403x2-8.738333333x+602.425.
After plotting the data and graphing the function over the data point one can see that the cubic Pt=-0.022466667x3+1.403x2-8.738333333x+602.425 fits most of the data points but not the first one or the last two. The graph is quite off from the last two points so to try and make the function fit better I replaced the last data point of (35, 1070) with (45, 1220.5) so that the function would have a stronger correlation with the data. The new function is Pt=-0.0141541667x3+1.0289375x2-3.958645833x+604.8390625. The revised function now reaches the top points though it does not line up with all points in is a better fit.
Next I used my TI-83 Plus to find the parameters K, L and M in the researchers model. I entered the original data into the Stat in my TI-83 Plus then performed a Logistic regression on the data. I then got an L value of 2.066777966, a M value of 0.0399301296 and a K value of 1617.459658. This would make the researchers model into Pt=1946.183708(1+2.619019043 e-0.0333213657x)
Year| Population in millions| Logistic Model| 0| 554.8| 537.77|
5| 609| 604.95|
10| 657.5| 676.5|
15| 729.2| 751.77|
20| 830.7| 829.94|
25| 927.8| 910.04|
30| 998.9| 991.01|
35| 1070| 1071.7|
40| 1155.3| 1151.1|
45| 1220.5| 1228.1|
Original Data vs. Researchers Model
I graphed the population in millions and the logistic model on the same graph so that I could see...