Find an expression for the function whose graph is the given curve.
The top half of the circle x^2+(y-4)^2=4
(y - 4)^2 = 4 - x^2
y - 4 = ±√(4 - x^2)
but we only want the top part, so
y - 4 = +√(4 - x^2)
y = √(4-x^2) + 4
Wolfram verification:
http://www.wolframalpha.com/input/?i=plot+y+%3D+%E2%88%9A%284-x%5E2%29+%2B+4+from+-2+to+2
To find an expression for the function whose graph is the top half of the circle x^2 + (y - 4)^2 = 4, we need to express y in terms of x.
First, let's rewrite the equation in simplified form:
x^2 + (y - 4)^2 = 4
Expand the square term:
x^2 + y^2 - 8y + 16 = 4
Move the constant term to the other side:
x^2 + y^2 - 8y + 12 = 0
Now, complete the square for the y-terms:
x^2 + (y^2 - 8y + 16) + 12 - 16 = 0
Simplify:
x^2 + (y - 4)^2 - 4 = 0
Rearrange:
(y - 4)^2 = 4 - x^2
Take the square root of both sides:
y - 4 = ± √(4 - x^2)
Rearrange again:
y = 4 ± √(4 - x^2)
Since the curve represents the top half of the circle, we take the positive square root:
y = 4 + √(4 - x^2)
Therefore, the expression for the function is y = 4 + √(4 - x^2).