how do you determine when the y is the first i mean you could technically have done this equation either way by simply putting the y setup in front of the x what makes it x first vs y first

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When the x is first the hyperbola opens side to side. The transverse axis connecting the foci is horizontal (x-1)^2/4 - (y+3)^2/9 = 1   When the y is first the hyperbola opens up and down. The transverse axis connecting the foci is vertical. (y+3)^2/9 - (x-2)^2/4=1