Let the graphic of our function be #Psi_f# with #f(x) = 4x^3-6x#, #f:RR->RR#.
There are two possible situations; the graphic is either symmetric with respect to the axis #x=m# or to a point #P(a,b)#.
These two have different definitions, as follow:
A graph #G_f# is symmetric to an axis #x=m# if and only if, from a point on #G_f#, say #A(x_A,y_A)#, the point symmetric to #A(x_A,y_A)# with respect to the axis #x=m# lies on that graph.
We can see a visual representation of this for parabolas:
graph{(y-x^2)((x-2)^2+(y-4)^2-0.03)((x+2)^2+(y-4)^2-0.03)=0 [-10.005, 9.995, -0.8, 9.2]}
The axis of symmetry for this graph (#y=x^2#) is #x=0#.
Basically, in other words, if #x=m# is an axis of symmetry for the graph of the function #f#, then
#f(x) = f(2m-x)#, #forallx in RR#
Before trying it out, let's define what the center of symmetry is.
A point #P(a,b)# is the center of symmetry to a graphic #G_f# if and only if, for a point #A# situated on #G_f#, the point situated the same distance away from point #P# and on the same line as #A# and #P# is also a point of the graph.
We define the mathematical relation to show this the following:
#f(2a-x)+f(x)=2b#, #forall x in RR => f(2a-x)=2b-f(x)#
While we could try this out for both relations and do lots of calculations, one simpler way is to observe that the function #x|-> 4x^3-6x# is odd, as showed below:
#f(x)=4x^3-6x#
#f(-x)=-4x^3+6x#
#:. f(-x)=-f(x)#
This slightly resembles our second relation, doesn't it? If we put them side to side, we can see the similarity:
#f(2a color(red)(-x))=2bcolor(red)(-f(x))#
#f(color(Red)(-x))=color(red)(-f(x))#
In order for these to be the same, we must have #a=b=0#.
Thus, for an odd function #f#, the graphic of the function #G_f# is symmetric to the point #P(0,0)#, which is just the origin, #O#.
Since the function we have, #f(x)=4x^3-6x#, is odd, it means that its graph #Psi_f# is #color(red)("symmetric with respect to the origin")# of the system, #O(0,0)#.
Bonus: If #g# is an even function, then the graphic of #g# is symmetric with respect to the axis #x=0#, which is the #y#-axis.
