How do you determine whether the graph of #y^2+3x=0# is symmetric with respect to the x axis, y axis or neither?
Equation is symmetric w.r.t.
graph{y^2+3x=0 [15, 5, 5, 5]}
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To determine the symmetry of the graph of ( y^2 + 3x = 0 ) with respect to the xaxis, yaxis, or neither:

With respect to the xaxis: Substitute ( y ) for ( y ) in the equation. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the xaxis.

With respect to the yaxis: Substitute ( x ) for ( x ) in the equation. If the resulting equation is equivalent to the original equation, the graph is symmetric with respect to the yaxis.

With respect to neither axis: If neither of the above substitutions results in an equivalent equation, then the graph is not symmetric with respect to either axis.
Performing the substitutions:

Substitute ( y ) for ( y ): ( (y)^2 + 3x = 0 ) Simplifying, we get: ( y^2 + 3x = 0 ) Since this is the same as the original equation, the graph is symmetric with respect to the xaxis.

Substitute ( x ) for ( x ): ( y^2 + 3(x) = 0 ) Simplifying, we get: ( y^2  3x = 0 ) This equation is not equivalent to the original, so the graph is not symmetric with respect to the yaxis.
Therefore, the graph of ( y^2 + 3x = 0 ) is symmetric with respect to the xaxis, but not the yaxis.
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