Circle equation vs. semicircle equation

[Lo.Lee.Ta.]

This may sound like a dumb question, but...

For the equation of a circle with a radius of 1, the equation is:

x^2 + y^2 = 1


But if we rearrange that to be: y= √(1 - x^2),
then it's only a semicircle...

Why is that? Why is it now a semicircle just because it got rearranged?

Thanks!

 

[Mark44]

This may sound like a dumb question, but...

For the equation of a circle with a radius of 1, the equation is:

x^2 + y^2 = 1


But if we rearrange that to be: y= √(1 - x^2),
then it's only a semicircle...

Your two equations are not equivalent, meaning that the first equation has more solutions than the second one.

If you solve for y in the first equation, you should get y = ±√(1 - x²). The pos. square root represents the upper half circle; the neg. square root represents the lower half circle.

Why is that? Why is it now a semicircle just because it got rearranged?

Thanks!