Translating sine and cosine

[tsunade33]

I'm having trouble translating a cosine function. The word problem is this: "A buoy bobs a total of 12 feet in 14 seconds to get back to its starting point. The average height of the top of the buoy is 15 feet above sea level. Write the equation with the maximum height at time = 0." I determined the basic equation for this to be f(x)=3cos((?/7)x) + 12, which I believe is correct.

However, I can't find out how to translate this equation, or do a phase shift, which is required in the second part of the problem: "Write an equation for the buoy's height with respect to time if its initial position is at its lowest point 80 seconds after midnight." What I have so far is f(x)=3cos((?/7)x+?)+12. I believe that the question is asking me to translate the graph so that it intersects the point (80,9) but I cannot seem to figure it out. Help would be appreciated.

 

[mmm4444bot]

The average height of the … buoy is 15 feet above sea level.

f(x)=3cos((?/7)x) + 12

Except for the constant 12, your definition of f(x) looks good, to me.

The statement above makes me think that the cosine curve needs to oscillate across the line y = 15.

In other words, the top of the buoy is moving up and down between 12 and 18 feet above sea level, so its "average" height in halfway inbetween.

f(x) = 3cos(?/7*x) + 15

?


"Write an equation for the buoy's height with respect to time if its initial position is at its lowest point 80 seconds after midnight."

I believe that the question is asking me to translate the graph so that it intersects the point (80,9)

I agree, although my interpretation above gives the coordinates as (80, 12).

I'm not sure why they say "initial" position, if x is defined as the number of seconds after midnight.

The period of f(x) is 14 seconds, and the lowest height occurs at 7 seconds.

It follows that the buoy will be at its lowest height in the middle of every consecutive 14-second interval. In other words, at the following values for x:

7, 21, 35, 49, 63, 77, 91 ?…

You need to create a new function g(x) such that the low occurs at x = 80, instead of 77. This means that you need to shift the graph of function f three seconds to the right, yes?

Do you remember what needs to be done to x, in order to shift a graph 3 units to the right?

 

[tsunade33]

After a bit of googling I found the equation phase shift = -c/b, which I had forgotten. After I found that, I was able to get the equation f(x) = 3cos((?/7)x - 3?/7) + 12, figuring I had to multiply the phase shift (+ 3) by b (?/7), and change the sign. So, I believe that my c of - 3?/7 is correct.

However, you have alerted me to another problem with my equation; the d value, or vertical shift. I see that your interepretation seems to make more sense than what I originally took to be "top" = maximum height. I will change the d to 15. Thank you for your help.