Plot[{Sqrt[x^3+5], -Sqrt[x^3+5]},{x,-3,3}]

Plot[{Sqrt[x^3-2], -Sqrt[x^3-2]},{x,-1,5}]

Show[Plot[{Sqrt[x^3+5], -Sqrt[x^3+5]},{x,-3,4}],ListPlot[{{-1,2},{41/16, 299/64}}]]

xcoord = ((#[[1]]^4 - 8*5*#[[1]])/(4*(#[[2]]^2)))&

ycoord = ((-#[[1]]^6 - 20*5*#[[1]]^3+8*5^2)/(8*#[[2]]^3))&

NestList[xcoord,{-1,2},1]

NestList[ycoord,{-1,2},1]

PointsOnCurve = NestList[{((#[[1]]^4 - 8*5*#[[1]])/(4*(#[[2]]^2))),((-#[[1]]^6 - 20*5*#[[1]]^3+8*5^2)/(8*#[[2]]^3))}&,{-1,2},4]

ListPlot[PointsOnCurve]

Show[Plot[{Sqrt[x^3+5], -Sqrt[x^3+5]},{x,-3,3}],ListPlot[PointsOnCurve]]

Show[Plot[{Sqrt[x^3+5], -Sqrt[x^3+5]},{x,-3,3}],ListLinePlot[PointsOnCurve]]