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Find an expression for the function whose graph is the given curve.

The top half of the circle $ x^2 + (y - 2)^2 = 4 $

$y=2+\sqrt{4-x^{2}}$

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okay. We want to find a function that represents the top half of this circle. And if you're familiar with the equations of circles, you probably recognize that this one happens to be a circle that shifted up two units. Its center is at 02 and its radius is two. So it looks something like this. And so what's gonna happen is we're going to solve this equation for why, and we're going to get something with a plus or minus in it. And the plus represents the top half on the minus represents the bottom half of the circle, so let's see what happens. So let's take our equation and first subtract X squared from both sides, and we get why minus two quantity squared equals four minus X squared. And then we're going to square root both sides. Our goal is to eventually have Why isolated. So when we square both sides, we have plus or minus the square root of four minus x squared, and then we're going to add to to both sides sweet of y equals two plus or minus the square root of four minus X, where so remember as I said before, because of the plus minus the pluses. The top half of the circle, the minuses, the bottom half. And we want the top half. So we want why equals two plus the square root of four minus X squared.