Period !
Time : 38s
Length of swimming pool: 25 m
So this weekend, i hoped into a pool and swam. My personal average time for a 50 meters freestyle is 38 seconds. The length of the pool is 25 meters. On the left diagram, you can see me dive in and swam 50 meters. The question here is how can swimming form a sinusoidal graph, the answer to that is swimmers swim in a periodical path - back and forth, on a graph this would be - Minimum and Maximum value.
Now we have a sense that the swimmer is swimming in a periodical path, what is the period then? The period is the time that the swimmer takes to complete one cycle, in my case - I completed one cycle (50 meters) in 38 seconds. Therefore the period is 38 in my case.
The reason that the my period is in radiant measurement is important to mathematics, especially trigonometry and calculus, in the sense that it provides a "pure," or unit-less measurement of an angle. This allows for very simple expression of derivative and integral relations that involve trigonometric functions in calculus.
HERE IS ONE ASSUMPTION I AM MAKING AND THIS IS IMPORTANT!
I am assuming that the swimmer is swimming forever and ever in a period of 38 seconds. This will help with forming my sinusoidal graph and equation
Length of swimming pool: 25 m
So this weekend, i hoped into a pool and swam. My personal average time for a 50 meters freestyle is 38 seconds. The length of the pool is 25 meters. On the left diagram, you can see me dive in and swam 50 meters. The question here is how can swimming form a sinusoidal graph, the answer to that is swimmers swim in a periodical path - back and forth, on a graph this would be - Minimum and Maximum value.
Now we have a sense that the swimmer is swimming in a periodical path, what is the period then? The period is the time that the swimmer takes to complete one cycle, in my case - I completed one cycle (50 meters) in 38 seconds. Therefore the period is 38 in my case.
The reason that the my period is in radiant measurement is important to mathematics, especially trigonometry and calculus, in the sense that it provides a "pure," or unit-less measurement of an angle. This allows for very simple expression of derivative and integral relations that involve trigonometric functions in calculus.
HERE IS ONE ASSUMPTION I AM MAKING AND THIS IS IMPORTANT!
I am assuming that the swimmer is swimming forever and ever in a period of 38 seconds. This will help with forming my sinusoidal graph and equation
Max & Min
These two important elements (period and the length of the pool) determined my sinusoidal function. In this graph, it clearly shows that I started swimming at the time and distance of 0, [on graph (0,0)] and i finished 50 meters in 38 seconds. The maximum point of this graph is (19, 25) which means that I reached the other side of the pool in 19 seconds. Since the swimming pool is only 25 meters in length, I have to go back to finish my other 25 meters. The minimum point is (38, 0) meaning that i have finished the 50 meters in 38 seconds and i came back to the point where I originally started which is zero.
The curve between the start (0,0) to the maximum point (19,25) is increasing, because the swimmer is swimming towards the other end of the swimming pool. The curve between the maximum point and minimum point is decreasing because the swimmer is swimming towards the other end of the pool.
The curve between the start (0,0) to the maximum point (19,25) is increasing, because the swimmer is swimming towards the other end of the swimming pool. The curve between the maximum point and minimum point is decreasing because the swimmer is swimming towards the other end of the pool.
What is the equation and transformation? and why?
This equation is an ideological equation for my average swimming record. f(t) is the distance I swam, and t is the time that i took to swim.
I have selected cosine functions as my parent function because the cosine parent function has a minimum of 1, this makes the transformations cleaner and easier to understand.
The amplitude is 12.5, means that from the maximum point or the minimum point to the mid-line of the graph is 12.5. In terms of swimming, this simply means that the swimmer is half way to the end of the pool since the length of the pool is 25 meters.
I have selected cosine functions as my parent function because the cosine parent function has a minimum of 1, this makes the transformations cleaner and easier to understand.
The amplitude is 12.5, means that from the maximum point or the minimum point to the mid-line of the graph is 12.5. In terms of swimming, this simply means that the swimmer is half way to the end of the pool since the length of the pool is 25 meters.
The period of this graph is 38 seconds. In the equation is written as
2π/38 (pi/16) because the period value is divided by 2π (which was the original value). In terms of swimming, the period means that the spent 38 second to finish one cycle - 50 meters. I went from 0 to 25 meters (the other end of the pool) and came back and 38 seconds is how long it took me to swim 50 meters. |
Reflection on the X-axis makes the the graph flip around. Therefore the maximum point at the start now has become the minimum point. This means the the swimmer starts at the minimum point and goes to the end of the pool which is the maximum point and comes back from the maximum point to the minimum point.
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Vertical translation is moved up 12.5 units. This is the last transformation, and the vertical translation makes the equation starts at zero. In terms of swimming, this means the swimmer starts at 0 meters and 0 seconds which is more reasonable than the swimmer starts at a negative distance.
Lastly, the restrictions(Limitation) of this equation is all real numbers of y is bigger and equal to zero, since there can not be a negative distance value. And all real numbers of x is bigger and equal to zero, since there cannot be a negative time value.
Lastly, the restrictions(Limitation) of this equation is all real numbers of y is bigger and equal to zero, since there can not be a negative distance value. And all real numbers of x is bigger and equal to zero, since there cannot be a negative time value.