Show the locus of the vertex of parabola

For the parabola y = a x^2+ b x + c I want to see the locus of the vertex as b varies (with a and c to be parameters). What are the ways to plot the locus of the parabola dynamically in Mathematica?

I am very new to Mathematica, so I could figure out only basic Manipulate:

Manipulate[
Plot[a x^2 + b x + c, {x, -10, 10}, PlotRange -> 40, AspectRatio -> 1],
{a, -20, 20}, {b, -20, 20}, {c, -10, 10}
]


I would like to see the locus parabola appearing in the same manner as in the post, if possible.

Thanks.

• Can you give more details about exactly what you want? As it is, this is likely to be closed as being unclear what you're asking. Mar 25, 2016 at 17:43
• I want something like this. A animation where the locus of the vertex of the parabola is plotted as i manipulate a,b,c. Mar 26, 2016 at 1:59

Manipulate[
plot1 = Plot[a x^2 + b x + c, {x, -10, 10}, PlotRange -> 40,
AspectRatio -> 1, ImageSize -> Small];
plot3 = If[a == 0, Graphics[],
ParametricPlot[{-u / (2 a), a (-u / (2 a))^2 + u (-u / (2 a)) + c}, {u, -b, b},
PlotStyle -> {Red, Thick}]];

Show[plot1, plot3],

{{a, -2}, -20, 20}, {b, -20, 20}, {c, -10, 10},
TrackedSymbols :> True]


Edit Another method to visualize locus:

Manipulate[
f[x_, i_] := a x^2 + (b + i) x + c;
tmp = Table[f[x, i], {i, -20, 20, 5}];

plot1 = Plot[tmp, {x, -10, 10}, PlotRange -> 40, AspectRatio -> 1];
plot2 = Plot[Tooltip[-a x^2 + c, "Locus"], {x, -10, 10}, PlotStyle -> {Black, Thick}];

Show[plot1, plot2],

{a, -20, 20}, {b, -20, 20}, {c, -10, 10}, TrackedSymbols :> True]


Edit2 Note on connection between first and second parts. Notice in 'plot3' we have used parametric form.

Solve[{x == -b / (2 a), y == a x^2 + b x + c}, {x, y}] // FullSimplify


$y=c-\frac{b^2}{4 a}$ and $x^2=\frac{b^2}{4 a^2}$.

So, we may come back from parametric to cartesian equation $y= - a x^2 + c$

(this equation is used in plot2).

• Thank you. And how to find the equation of that locus? Mar 26, 2016 at 15:52
• @LokeshJaddu, in the second visualisation it is -a x^2 + c. In practice it is more complicated, but in Manipulate your a and c change sign from - to +, so we may use this very simple representation. Mar 26, 2016 at 16:03
• I understood that its -a x^2 + c . But isn't there a way to let mathematica figure that equation out for me? Mar 26, 2016 at 16:08
• You can use GroebnerBasis[] for eliminating parameters. Mar 27, 2016 at 0:43