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Question 1140835: Here is a link to my problem. I have tried to solve it but I just don't understand.
https://gyazo.com/c627cecf56b8cc8e0b2b9640f0149b73
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i was able to crate the graph using desmos.com graphing calculator.
the equation is y = 2 * cosine (90 * x) + 1
the general form of the equation is:
y = a * cosine (b * (x - c)) + d
a is the amplitude
b is the frequency
c is the horizontal displacement
d is the vertical displacement.
your equation is, once again, y = 2 * cosine (90 * x) + 1
the amplitude is 2 which means plus or minus 2 units from the horizontal center line.
the frequency is 90 because the periods needs to be 4 and the equation for frequency is that the frequency is equal to 360 degrees / period.
since the period is 4, the equation becomes frequency = 360 / 4 = 90.
the period is the length of one full cycle of the function.
the function is cosine.
cosine (0) = 1 if you don't multiply it by an amplitude of greater than 1.
one full cycle of the cosine wave goes from y = 1 to 1.
when the amplitude is 2, one full cycle goes from y = 2 to 2.
since the amplitude is plus or minus 2 units from the center line, the amplitude would go from -2 to 2 if there were no vertical displacement.
there is no horizontal displacement so c = 0.
the vertical displacement is 1.
this makes the center line of the function equal to y = 1 rather than y = 0 where it would be if there were no vertical displacement.
the bottom line is that the equation y = 2 * cosine (90 * x) + 1 has the following properties.
the frequency is 90 because frequency = 360 / period and the period needed to be 4, based on the diagram.
the center line of the function is at y = 1 because the function has been vertically displaced up one unit.
the amplitude is plus or minus 2 from the center line.
this makes it go from -1 to 3 because the center line is at y = 1.
the answers to your questions are;
the period of one complete cycle is 4.
the range of the function is y = -1 to y = 3
if the particle can survive for twenty complete cycles before deteriorating, then the domain of the function would be 0 <= x <= 80.
this is because one full cycle is 4 which makes 20 full cycles = 4 * 20 = 80.
the equation of the horizontal axis is y = 1 because the function has been vertically displaced up one unit.
this is because the cosine wave is plus or minus 2 units from the center line.
since the diagram shows the range of y going from -1 to 3, the center line had to be displaced up one unit so the cosine wave was symmetric about the center line, or what could be called the horizontal axis of the function.
amplitude is 2 because the spread between the highest level and the lowest level is 4, which makes the amplitude plus or minus 2 from the center line.
my graph of the equation is shown blow.
the first graph is one full cycle.
the second graph is 20 full cycles.
any questions, write to dtheophilis@gmail.com
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