felpTide Modeling

In order to come up with a plot graph for the bay of Fundy in Noca Scotia, Cana it was necessary to use Microsoft Excel and the data from . In order to plot the graph in excel all of the data had to be entered in the spreadsheet.

Once that was done a scatter graph was created. Once it was done it was possible to analyze the graph. The first thing that is possible to notice is that this is a periodical graph. However, the graph is not completely periodical since the lines don’t always have the same height. Sometimes it is greater or lower. This can be attributed to the fact that it is a real life situation. Therefore, it would not be expected for it to be perfect. However, the graph is still periodical since it follows the same shape throughout.

In order to come up with the equation I had to come up with a series of averages. This was necessary since the graph is not completely periodical.

In order for me to find the vertical translation of the graph, meaning how much the graph was moved up I had to find the graph’s sinusoidal axis. In order to find it I had to find the graphs height for both crests and do an average of it. The highest point in the first crest was 12 meters. For the second crest it was 12.3 meters. Therefore, my average was 12.15 meters. My next step was to find the lowest points, the trough. For the first one was 0.90 meters and for second one it was 0.70. Therefore, my average was 0.80 meters. Finding the average between the through and the crest gave me the sinusoidal axis. The sinusoidal axis ended up being 6.475 meters.

Calculations: Vertical Translation d= 6.475

12+12.3=24.3

,24.3-2.=12.15

0.90+0.70=1.6

,1.6-2.=1.6

0.8+12.15=12.95

,12.95-2.=6.475

My next step was to find the graphs vertical stretch also known as the graphs amplitude; or the distance between the sinusoidal axis and the crest. In order to determine, it was necessary to subtract the sinusoidal axis from the average crest. My answer for the vertical stretch was than 5.675.

Calculations: Vertical Stretch a= 5.675

12.15−6.475=5.675

The next step was to find the graphs horizontal dilation also known as the graphs period. In order to determine it is necessary to find two points exactly on the same y coordinate and different x axis. It is also important to notice that they need to be in the same position of the period. For reference it was measure the top two points of the crests, and the two low points of the trough. Once the two points are measure it is necessary to find the difference between the two points. In this case the difference was 12. Once it is found it is needed to plug it in the formula( 𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒=,2𝜋-𝑏.), and solve it for b. That value will be the graphs period. In this case after using the formula the answer that was acquired was,𝜋-6..

Horizontal Dilation b=,𝜋-6..

15−3=12 Crest

21−9=12 Through

12=,2𝜋-𝑏.

𝑏=12

Finally, the last step in order to determine a model for the graph is its horizontal translation. In order to do the last step, it is necessary to determine if the graph is cosine or sine.

First it will be assumed it is cosine. Once that is done it is necessary to calculate how much the graph was moved from its original position. In order to determine it, the difference between the standard cosine graph and the translated one needs to be found. In order to do so two of the same points need to be analyzed for example two crests. Once the comparison is made it is possible to see that the graph was moved 3 units to the right.

Horizontal Translation c=3

Since the crest in the standard graph was at 0 and in the excel graph it is at 3 it is possible to assume it was moved 3 units to the right.

All of the vales above can also be used for sine since cosine and sine are only a horizontal translation of each other. No transformation would be required because as the graph is half way up it crosses the y- axis and it also happens in the transformed.

Horizontal Translation c=0

Therefore the following models are possible to come up with after the analysis:

Cosine:

𝑦=5.675,cos-,,𝜋-6.,𝜃−3...+ 6.475

Sine:

𝑦=5.675,sin-,,𝜋-6.,𝜃...+6.475

The graph to the side further proves that this is an accurate model because it shows that the model goes through most of the points of the original excel graph.