(x,y) and (-x,y) are symmetric w.r.t. the Y-axis.
(x,y) and (x,-y) are symmetric w.r.t. the X-axis.
(x,y) and (-x,-y) are symmetric about the origin.
Symmetries of the graph of an equation (or restricted to a function):
Y-axis symmetry: 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: 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: every (x,y) iff (-x,-y). ≡ 180° rotation about origin.
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).
(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.
If the graph is neither X nor Y symmetric,
Test for Origin symmetry: replace x by -x and y by -y.
For a function: if ƒ is not even, test if -f(x) = f(-x) then ƒ is symmetric about origin;
it is an odd function.
Also, x=y main diagonal line symmetry: every (x,y) iff (y,x)
See inverse functions
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. |