$x^2+y^2=25$ vs $y=\sqrt{25-x^2}$

Question

Why does every graphing calculator give me a graph with a circle when I give it $x^2+y^2=25$ (circle equation), but only half a circle when I give it $y=\sqrt{25-x^2}$ (same, only solved for $y$)?

Cheers.

Answer 1

when you solve the equation $$x^2+y^2=25$$ for $y$ you must write $$y=\pm\sqrt{25-x^2}$$

Answer 2

Because $x^2+y^2=25$ allows negative values of $y$ too, while the equation $y=\sqrt{25-x^2}$ doesn't.(Since $\sqrt{a}$ is always positive, $\forall ~a>0$ )

Answer 3

Common misunderstanding for beginners. The following should clear up the confusion.

1. It's ok to say in words that the square root of $9$ is $\pm3$.
2. But when using $\sqrt{9}$ notation, one has to be careful. That notation denotes the non-negative value of the square root, namely, writing $\sqrt9$ means $+3$. It's a convention.
3. Thus, to indicate there are two roots, one has to write $\pm3$ or $\pm\sqrt{9}$ because $\sqrt9$ simply means $3$.

In general, if the square root is a real number, the convention is that the notation $\sqrt{...}$ denotes a non-negative value:

$$ \sqrt{x^2} = |x|.$$

Again, this convention does not apply to complex numbers.