Find an expression for the function. Top half of the circle: x^2+(y-2)^2=4.
(y-2)^2 = 4 - x^2
y-2 = sqrt(4-x^2)
y = sqrt(4-x^2) + 2
To find an expression for the top half of the circle defined by the equation x^2 + (y - 2)^2 = 4, we need to solve the equation for y.
Let's start by simplifying the equation:
x^2 + (y - 2)^2 = 4
Expand the square:
x^2 + y^2 - 4y + 4 = 4
Subtract 4 from both sides:
x^2 + y^2 - 4y = 0
Rearrange the terms:
y^2 - 4y = -x^2
Complete the square by adding (4/2)^2 to both sides:
y^2 - 4y + 4 = -x^2 + 4
Factor the left side:
(y - 2)^2 = -x^2 + 4
Now, take the square root of both sides, but since we are interested in the top half of the circle, we only consider the positive square root:
y - 2 = √(-x^2 + 4)
Simplify:
y = 2 + √(-x^2 + 4)
Therefore, the expression for the top half of the circle is y = 2 + √(-x^2 + 4).
To find the equation of the top half of the circle, we need to solve the given equation for y.
Step 1: Start with the equation of the full circle:
x^2 + (y - 2)^2 = 4
Step 2: Expand the equation:
x^2 + y^2 - 4y + 4 = 4
Step 3: Group the terms containing y together:
y^2 - 4y = 4 - x^2
Step 4: Complete the square on the left side of the equation:
(y - 2)^2 = 4 - x^2
Step 5: Take the square root of both sides:
y - 2 = ± √(4 - x^2)
Step 6: Solve for y by adding 2 to both sides:
y = 2 ± √(4 - x^2)
Since we are looking for the top half of the circle, we only consider the positive square root. Therefore, the expression for the function representing the top half of the circle is:
y = 2 + √(4 - x^2)