Question 687196
a) {{{(x-2)^2+y^2=4}}}

{{{ graph( 600, 600, -10, 10, -10, 10, sqrt(-(x-2)^2+4),-sqrt(-(x-2)^2+4)) }}}

It's the one that has y^2 in it because you could have one x value with two different y values ( a positive and a negative, since y^2 is positive for both).

If you you use only {{{y = sqrt(-(x-2)^2+4)}}} you wold have a {{{semicircle}}}, which is a {{{function}}}.

On a graph, the idea of single valued means that no vertical line would ever cross more than one value.

If it crosses more than once it is still a valid curve, but it would not be a function.

{{{drawing(600,600,   -6, 6, -6, 6,  blue(line(2,6,2,-6)), grid(0),
graph( 600, 600, -6, 6, -6, 6,sqrt(-(x-2)^2+4),-sqrt(-(x-2)^2+4)))) }}}

b) {{{x^2+4x+y=4}}}

{{{ graph( 600, 600, -10, 10, -10, 10, x^2+4x-4) }}}

it is a function

c) {{{x+y=4}}}

{{{ graph( 600, 600, -10, 10, -10, 10, -x+4) }}}

d) {{{xy=4}}}

{{{ graph( 600, 600, -10, 10, -10, 10, 4/x) }}}

this is hyperbola


not all hyperbolas are functions, but the one you have {{{IS}}} a function It can also be written as:

{{{y = 4/x }}}

the ones that are {{{NOT}}} functions are in the form of {{{x^2/a^2 - y^2/b^2 = 1}}}; whenever {{{y}}} is {{{squared}}}, you do {{{not}}}{{{ have}}} a {{{functions}}}