Modeling the Weather

The table shows Melbourne’s mean average daily maximum temperature (℃) for two year period 1999-2000.

Year| Jan| Feb| Mar| Apr| May| Jun| Jul| Aug| Sep| Oct| Nov| Dec| 1999| 25.7| 26.9| 24.5| 21.4| 18.0| 14.0| 13.5| 13.9| 17.2| 19.4| 22.2| 24.6| 2000| 26.0| 25.4| 24.7| 20.7| 17.5| 14.6| 14.8| 14.4| 17.5| 20.6| 22.9| 26.1|

1. Define appropriate variables and parameters, and identify any constraints for the data. Independent variable: months

Dependent variable: temperature

Range: 13.5≤y≤26.9

2. Use technology to plot the data points on a graph. Comment on any apparent trends shown in the graph. By plotting the data given from the table into graphmatica, the graph below can be obtained. GRAPH 1

Temperature (℃)

Months

This graph shows that if we connect these points, we will see a regular wavy curve appropriately.

3. What type of function models the behavior of the graph? Explain why you chose this function. I noticed that the sine graph best models the behavior of the graph. The reason is that GRAPH 1 simply looks like a sine function and it seems contain amplitude and period.

Sine function graph : y=sin(x)

4. Use your knowledge of the graphs of such functions to create a suitable equation that models the behavior of the data. Explain all steps you took to arrive at your equation. First of all, the general sine function is defined as: y=AsinB(x-C)+D * A represent the Amplitude

* B represent the n in Period (T=2πn)

* C represent the Horizontal translation

* D represent the Vertical translation

* To find A, I know the ymin=13.5 and ymax=26.9 from the table. Amplitude = ymax-ymin2 = 26.9-13.52 = 6.7

A is therefore 6.7.

* To find B, I know that the period (T)=12 months.

T=2π BT=12

B = 2π12 = π6

B is therefore π 6.

* To find D, I know the ymin=13.5 and ymax=26.9 from the table. D= ymax+ymin2 =...

(1)