Ok, we want to build up a sinusoidal function to have a maximum at t=6 seconds, a period of 10 seconds, a minimum height of -200 cm from the surface of the trampoline, and an unnamed maximum height we can just call H.
Since we're given information about an extreme of the graph (a max or min) instead of about the midline, I think it's easier to base everything off cosine, since when we haven't transformed the cos function at all its maximum is nicely placed at (0,1).
Whenever I'm dealing with the inputs of a sinusoidal function, I like to think about it in terms of units: trig functions are normally expecting to be given something in radians, whereas in this situation we'd like our horizontal t-axis to represent something in seconds. So anytime we think about putting something into cos here that was originally in our t-axis units of seconds, we want to convert from seconds to radians.
A common reference point of seconds vs. radians here is how long does one repetition of the graph take? From peak to peak of the cosine graph is normally 2*pi radians; in this situation, we also know it's 10 seconds. This tells us we can convert between seconds and radians using this proportion!
In particular, given something in seconds, we can multiply by 2*pi radians/10 seconds - essentially multiplying by 1, since those are the same thing expressed in different units - and this lets us cancel out seconds with seconds and get something in the radians that cos wants.
Our input begins to look like 2pi/10 * t
We furthermore want a horizontal translation of the graph 6 units to the right; essentially, we want when we plug 6 in, for it to act like we've plugged in 0 into the original cosine function to give us that maximum value.
I see folks get confused about whether our input should thus be (2*pi/10 * t) - 6 or 2*pi/10 * (t - 6). Easy way to keep track of what's right: are we talking about shifting by 6 radians, or 6 seconds? Since we want the peak of the graph to be at 6 seconds, we want to multiply the 6 by this unit-converting 2*pi/10 factor as well: our input should be
2*pi/10 * (t-6).
vs. if we instead were in a situation where we wanted to shift 6 radians, then we would not group the 6 with the t, since it would already be in radians & not need an additional unit conversion.
Cool, we've got our input to cosine of 2*pi/10 * (t-6).
Now we just need to adjust for the amplitude and midline of the graph.
Owen's jumps take him from a minimum of -200 cm to an unknown max of H - let's also let this be in cm - , for a total of H+200 cm. The normal total range of a sinusoidal function is from -1 radius length to +1 radius length, for a total of 2 radius lengths. We can set up a similar conversion process for our output, where cosine puts out radius lengths but we'd like in this case for it to put out Owen's height off the trampoline in cm, so we can multiply by (H+200 cm)/(2 radius lengths) to get the right amplitude (vertical stretch) of our function.
We've right now got something that looks like (H+200)/2 * cos(2*pi/10 * (t-6)). Right now, this graph is centered at a height of 0, so our final step will be to add the height Owen is at mid-bounce.
With a minimum of -200 cm and a maximum of H cm, the middle of these heights is (-200+H)/2. We add this to the end to get the correct midline, and our final answer of:
Owen's height as a function of time can be described by (H+200)/2 * cos(2*pi/10 * (t-6)) + (-200+H)/2
There's definitely some small simplifications we can do here, like recognizing 2*pi/10 = pi/5, but these aren't necessary for modelling the function, just for showing off your algebra skills. Anything algebraically equivalent to this answer will give the same graph, and so still be a correct model for the situation.
I hope this is helpful! If you have any follow-up questions - including if you have additional information which might help us understand Owen's maximum height - please feel free to leave a comment, I'd be glad to clear up any lingering confusion.
