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Sometimes a phenomenon can be periodic even though it is not connected to motion in a circle. For example, here is a graph of the water level in Bridgeport, Connecticut, over a 50-hour period in 2018.
Graph of a function, origin O. Horizontal axis from 0 to 50, by 10s, time, hours after September 1st 2018. Vertical axis from 0 to 8, by 2s, water level, feet. Function begins on the y axis at 2, decreases to about 2 point 5 comma 1, then increases to about 7 point 5 comma 7 point 5, decreases to about 14 comma 0 point 8, increases to 20 comma 8, decreases to about 27 comma 0 point 8, increases to about 32 point 5 comma 7, decreases to about 38 comma 1, increases to about 45 comma 7 point 5, then decreases to about 50 comma 1 point 5.
The midline of the graph appears to be around 4.5 feet. Notice that each day (or each 24-hour period) there are two high tides, a small one where the water goes up by a little less than 3 feet and then a bigger one when the tide goes up by a little more than 3 feet. Since there are two high tides per day, the period for this graph is about 12 hours. The data begins about 1 hour before the tide is at the 4.5 midline value. Since , this would make the sine function a good choice for modeling the tide.
Putting together all of our information gives the model , where measures hours since midnight on September 1.
Graph of two sine functions, on a coordinate plane with grid, origin
. The functions are of the water level in Bridgeport, Connecticut, over a 50 hour period in 2018. Horizontal axis scale 0 to 50 by 10’s, labeled “time (hours after September 1, 2018)”. Vertical axis scale 0 to 8 by 2’s, labeled “water level (feet)”. Graph of the model equation
is blue and similar to the black function. The model function begins on the y axis above (0 comma 2), decreases to about (1 point 8 comma 1), then increases to about (7 comma 7), decreases to about (14 comma 1), increases to about (20 comma 7), decreases to about (26 comma 1), increases to about (32 comma 7), decreases to about (38 comma 1), increases to about (43 comma 7), and then decreases to about (50 comma 1).
Notice that: