- Thread starter apple2357
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The area in each case is half of a rectangle with base pi/2 and height 1, because the curve bisects that rectangle. And that happens because f(pi/2 - x) = 1 - f(x).

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I saw the answer purely visually; the last graph was most obvious, then I saw the symmetry in the others, and then I realized how to see that in the function itself. It's all about cofunctions.

If it was f(x-pi/2) i would understand that as a simple translation but is f(pi/2-x) is a combination of transformations? Is this a translation and a reflection?

How did you come up with f(pi/2-x) = 1-f(x) as the explanation for the graphical observation?

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Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!

So If we reflect y=f(x) in x=a, we get the curve y=f(−x+2a) ?

Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!

Is this because if you want to move every point to a position the same distance the other side of x=a we end up at x+2(a-x). Which gives us 2a -x

Is that how you would explain it?

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And, of course, the same fact is why the other reflection is 1 - y.