Symmetry
Symmetry: the graph will look the same on either part of an axis or boundary
x-axis symmetry: portion of graph above x-axis is the same as below
y-axis symmetry: portion on left of y-axis is the same as on left
origin symmetry: the graph would look the same if rotated 180 degrees about the origin (left figure)
y=x symmetry: the graph is symmetrical about the line y=x (the parent line graph)
Y-axis Symmetry: substitute -x for x in the equation and see if the equation remains the same (has the same value). If it does, then the equation has y-axis symmetry. Try: y = 2x
y = 2x
y = 2(-x)
y = -2x
The value of this equation has changed. Therefore, this equation does not have y-axis symmetry.
We could easily alter this equation to have x-axis symmetry, though. Let's square 2x: y = 2x². The function now has x-axis symmetry.
X-axis Symmetry: replace y with -y and if the value of the equation does not change, then the equation has x-axis symmetry. Try: y = x².
y = x²
-y = x²
y= -x²
This function has x-axis symmetry. You can always check your answer by looking at the graph on a graphing calculator.
Origin Symmetry: replace x and y with -x and -y respectively and see if the equation remains the same, if it does then the equation has origin symmetry. Try: y = 2x²+3
y = 2x²+3
-y = -2x²+3
y = 2x²-3
This equation has changed, so it does not have origin symmetry. (Shown on red graph below.)
But, try typing y = 1/x into your graphing calculator. That equation does have origin symmetry.
Symmetry y=x: (also known as diagonal symmetry) check for y = x symmetry by switching x and y (like we do to find inverses) and seeing if the equation remains the same. Try: y = 3x+4
y = 3x+4
x = 3y+4
x-4 = 3y
(x-4)/3 = y
This equation does not have y = x symmetry. (Shown on the orange graph below).
y = 2x
y = 2(-x)
y = -2x
The value of this equation has changed. Therefore, this equation does not have y-axis symmetry.
We could easily alter this equation to have x-axis symmetry, though. Let's square 2x: y = 2x². The function now has x-axis symmetry.
X-axis Symmetry: replace y with -y and if the value of the equation does not change, then the equation has x-axis symmetry. Try: y = x².
y = x²
-y = x²
y= -x²
This function has x-axis symmetry. You can always check your answer by looking at the graph on a graphing calculator.
Origin Symmetry: replace x and y with -x and -y respectively and see if the equation remains the same, if it does then the equation has origin symmetry. Try: y = 2x²+3
y = 2x²+3
-y = -2x²+3
y = 2x²-3
This equation has changed, so it does not have origin symmetry. (Shown on red graph below.)
But, try typing y = 1/x into your graphing calculator. That equation does have origin symmetry.
Symmetry y=x: (also known as diagonal symmetry) check for y = x symmetry by switching x and y (like we do to find inverses) and seeing if the equation remains the same. Try: y = 3x+4
y = 3x+4
x = 3y+4
x-4 = 3y
(x-4)/3 = y
This equation does not have y = x symmetry. (Shown on the orange graph below).