Question

why do x=sqrt(y) and y=x ^2 have different graphs?

Answer 1

The square root function works like this

you plug in some positive number and out pops a positive number

you can plug in 0 and you'll get 0 out

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so that means you cannot plug in negative numbers and you will not get negative numbers as outputs. This is assuming you have a positive square root

Answer 2

so if x = sqrt(y), then x is either 0 or some positive number

effectively x is restricted to the right half of the xy axis