Symmetry

Symmetric pair of points:

Y-axis symmetry: (x,y) and (-x,y) are symmetric w.r.t. the Y-axis. As are (-x,-y) and (x,-y). The x's are negatives of each other. They "jump"/"flip" across the Y axis. A horizontal line connects them. X-axis symmetry: (x,y) and (x,-y) are symmetric w.r.t. the X-axis. As are (-x,-y) and (-x,y). The y's are negatives of each other. They "jump"/"flip" across the X axis. A vertical line connects them. Origin symmetry: (x,y) and (-x,-y) are symmetric about the origin. As are (-x,y) and (x,-y). Both the x's and the y's are negatives of each other. They "jump"/"flip" across both the X axis and the Y axis, in either order. They are both X-axis symmetric and Y-axis symmetric. Equivalently, they rotate 180° around/about the origin. A line through the origin connects them.

Whichever symmetry, the pair of points are the same distance from the X-axis, the same distance from the Y-axis, and the same distance from the origin.

Symmetries of the graph of an equation (or restricted to a function):

Y-axis symmetry: flipping every point across the Y-axis gives the same graph.
Every (x,y) iff (-x,y).
A reflection across the Y axis. The mirror image looking at the mirror on the Y axis.
A horizontal line connects each pair of symmetric points.
    f(x)=f(-x) and -f(x)=-f(-x) i.e. Even function.

X-axis symmetry: flipping every point across the X-axis gives the same graph.
Every (x,y) iff (x,-y).
A reflection across the X axis. The mirror image looking at the mirror on the X axis.
A vertical line connects each pair of symmetric points.
NB. A function cannot be symmetric about the X-axis.
Multiplying a function by -1, i.e. -f(x), flips it over X axis.

Origin symmetry: flipping every point across both the X-axis and the Y-axis (in either order) or, equivalently, rotating every point 180° about origin gives the same graph.
Every (x,y) iff (-x,-y). ≡ .
A reflection across X axis and across Y axis, in either order.
A line through the origin connects each pair of symmetric points.
    f(-x)=-f(x) and f(x)=-f(-x) i.e. Odd function.

A graph can be none, one, or all three of these symmetries. i.e. 0, 1, or 3 of them.
If is X-axis symmetric and Y-axis symmetric, then is Origin symmetric: X & Y → Origin. (but Origin symmetry does not necessarily imply X and Y symmetry.)
If a graph is any two of these symmetries, it is also the third (i.e. can not be symmetric with only two of them).
The graph of a function can be one or none (either Y symmetry, or Origin symmetry, or neither).

Algebraic testing for symmetries:
(If not a function) Test the equation for X-axis symmetry: replace y by -y; if this new equation equals the original, the equation is X-axis symmetric.

Test for Y-axis symmetry: replace x by -x; if this new equation equals the original, the equation is Y-axis symmetric.
For a function: if f(x) = f(-x) then ƒ is symmetric about Y-axis; it is an even function; Stop.

If the graph is neither X nor Y symmetric, Test for Origin symmetry: replace x by -x and y by -y. (If the graph is both X and Y symmetric, it is also Origin symmetric.)
For a function: if ƒ is not even, test if -f(x) = f(-x) then ƒ is symmetric about origin (if the function is even, can't also be Origin); it is an odd function.

Summary:

Function: 
if Y? then 
  "Y" 
else 
  if O? then 
    "O" 
  else 
    "None"

Equation (relation):
if X? then
  if Y? then
    "XYO"
  else
    "X"
else
  if Y? then
    "Y"
  else
    if O? then
      "O"
    else
       "None"

f(x)	f(-x)	-f(x) [-y]	-f(-x)	type of symmetry
=	=			Y-axis. EVEN function
		=	=
	=	=		Origin. ODD function
=			=
2x	-2x	-2x	2x	Origin, Odd. Linear thru origin.
2x-2	-2x-2	-2x+2	2x+2	Neither. Linear not thru origin.
2x2-3	2x2-3	-2x2+3	-2x2+3	Y-axis, Even. Quadratic, no bx term.
2x2+2x-3	2x2-2x-3	-2x2-2x+3	-2x2+2x+3	Neither. Quadratic, with bx term.
x3	(-x)3	-x3	-(-x)3	Origin, Odd. Cubic, no other terms.
∛x	∛-x	-∛x	-∛-x	Origin, Odd.
ex	e-x	-ex	-e-x	Neither.
1/x	-1/x	-1/x	1/x	Origin, Odd.
1/x2	1/(-x)2	-1/x2	-1/(-x)2	Y-axis, Even.
sin x	sin -x	-sin x	-sin -x	Origin, Odd.
cos x	cos -x	-cos x	-cos -x	Y-axis, Even.

Another kind of symmetry: across the y=x main diagonal line.

(a,b) and (b,a). Interchange/swap the coordinates. The line connecting the points is perpendicular to the y=x line. It is y=-x+(a+b) The points are |a-b|/√2 from the y=x line. The points are √(a2+b2) from the origin. See inverse functions Self-inverse functions such as y=x, y=-x, y=1/x, y=-1/x, y=(x+1)/(x-1) have this symmetry. Graphs with this symmetry: Every (x,y) iff (y,x).    Circles centered on the y=x line.    Tilted parabolas whose axis is symmetry is y=x.    Tilted ellipses with either axis on the y=x line.