ClearAll[f];
f[F_, y_] = x /. Solve[x^2 == (x - F)^2 + y^2, x][[1]]
Manipulate[
 Show[
  ParametricPlot[{f[F, y], y}, {y, -6, 6}],

  Graphics[{
    Point[{f[F, y], y}], Text["M", {f[F, y], y} + 0.25],
    Point[{F, 0}], Text["F", {F, 0} + 0.25],
    Point[{0, y}], Text["A", {0, y} + 0.25],
    Line[{{0, y}, {f[F, y], y}}],
    Line[{{F, 0}, {f[F, y], y}}]

    }],
  Axes -> True, PlotRange -> {{-10, 10}, {-6, 6}},
  AxesOrigin -> {0, 0},
  ImageSize -> Large
  ],
 {{y, 1}, -6, 6}, {{F, 1}, -10, 10}
 ]

standardParabola[x_] := (x^2);
ListLinePlot[
 Table[Evaluate@{standardParabola[x], x}, {x, -5, 5, .01}]
 ]

Plot[{Sqrt[x], -Sqrt[x]}, {x, -2, 4},
 PlotStyle -> {Blue, Blue},
 Epilog -> {Green, Thickness[0.005], Line[{{-1, -2}, {-1, 2}}],
   Red, PointSize[0.02], Point[{1, 0}],
   Dashed, Line[{{1, 0}, {.25, .5}, {-1, .5}}]}]