Determining if an area is symmetrical about the y axis

[frog1944]

Hi,

In Eddie Woo's video on volumes he states how he can take advantage of the symmetry of the graph (https://youtu.be/k2EQqwmeKjc). How did he determine it was symmetrical? Like it definitely looks like it, but how would you mathematically show it?

Thanks

[GodlyOne]

I think If the graphs points of intersections (x coordinates) are equal on both sides (without the signs) or if its axis of symmetry is the axis of rotation.

Hope that helps

[RuiAce]

Exploiting symmetry is hard. Most of the time people just do it by inspection of the graph

The symmetry here was exploited by considering the fact that the y-axis is the axis of rotation, as mentioned. This gives us a reference point; we should use our graph as a visual aid to determine whether or not the graph would exhibit some kind of symmetric behaviour about x=0, and yes it does.

Indeed, the graph has rotational symmetry about the point (0,12). Recall that to show the graph exhibited rotational symmetry about (0,0), or equivalently an odd function, we used the check f(-x) = -f(x). Here, the check would be \(f(-x) = -f(x) + 24 \)

This is obviously well beyond the HSC course, and symmetry arguments can (albeit must be carefully done so) by just graphical aid. It is not obvious at all immediately why we should consider 24, which is \(2\times 12\), unless we think hard enough.