Manipulate[
Graphics[
{ 
Line[{{-10,0},{10,0}}],
Line[{{0,-10},{0,10}}],
{Red,
Inset["y =", {-8.3,8}, {1,.3}],
Inset[a x^2 + b x + c, {-8,8},{-1,0}],Line[Table[{x,a x^2 + b x +c}, {x, -10, 10,.01}]],
Dashed, 
Lighter[Red],
Line[{{-10,(-b^2+4a c)/(4a)},{10,(-b^2+4a c)/(4a)}}],
Line[{{-b/(2a),-10},{-b/(2a),10}}]
}, 
If[bVertexPath,{
Green,
Line[Table[{x,-a x^2 +c}, {x, -10, 10,.01}]]
},{}],
If[aVertexPath,{
Blue,
Line[Table[{x,(b/2)x+c}, {x, -10, 10,.01}]]
},{}]
},
PlotRange->{{-10,10},{-10,10}}
], 
{a,-10,2},
{b,-10,10},
{c, -2, 2},
{aVertexPath, {True, False}},
{bVertexPath, {True, False}}
]