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The average time difference between high tides is the period of the function. So \(\displaystyle \frac{2\pi}{b} = 12.4\). So

The value of d depends on how you define H.

If you let H = "height above "mid-tide" " then d=0. Here H=-7 at low tide.

Also. the value of c depends on where you are measuring time from.

If you measure time from mid-tide on a rising tide c=0.

Personally I would choose the bold options above. The equation would be: H = ???

The graph would have the equilibrium line at y=7 (ie d). It would begin at the maximum point (0, 14), cut the equilibrium line at(3.1, 7) have a minimum on the x-axis at (6.2, 0) up to the equilibrium line again at (9.3, 7) up to a max again at (12.4, 14).

Other answers would be possible depending on how H and t are defined ie where they are measured from.

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Isn't y = asin(bt-c)+d easier to work with?

One period of the sin graph starts when bt-c=0 or t = c/b

That period ends when bt-c =2pi or t = (2pi+c)/b

And the period is therefore 2pi/b

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Both are useful, and are easy to use once you learn the relevant formulas; but I personally prefer the factored form a sin(b(t - c)) + d, because the shift is explicitly given as c, rather than having to be extracted by dividing. I've taught from books that do it either way, and live with it.

Isn't y = asin(bt-c)+d easier to work with?

One period of the sin graph starts when bt-c=0 or t = c/b

That period ends when bt-c =2pi or t = (2pi+c)/b

And the period is therefore 2pi/b

In general, the form f(a(x - b)) represents a "stretch" by 1/a followed by a "shift" of b; f(ax - b) represents a shift by b followed by a stretch by 1/a which modifies the shift to b/a, so that the shift you are doing is not the shift you see in the graph.

Actually, now that I think about it, if I were writing the book, I might use f((x - b)/a), so that x is multiplied by a and then increased by b. Then I could point out that u = (x - b)/a is the inverse of x = au + b, and a lot of things fall into place (specifically, why things work backward). But I don't want to get too far out of step with what all the textbooks say. Then, in the case of the sine, we'd have y= a sin((t - c)/b) + d, where the period is b times 2 pi and the horizontal shift is c, just as the amplitude is a and the vertical shift is d.

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Using sines, the easiest thing to do is to start from the point between high and low (mid-tide), since then no phase shift is needed. For practical purposes, you would want to start at midnight; but we don't have that information. In light of that, I would tend to start from low or high tide, which are more natural times to observe; but then I would naturally use a cosine model.

So who are you taking as the authority on what you "should" do?