A graph is said to have symmetry if it can divided into two equal parts along a line such that one part is an exact mirror image of the other. The concept is useful to graph an equation.
Types
Graph symmetry is of 3 types:
Graph Symmetry
x-axis Symmetry
A graph is symmetric about the x-axis when the points (x, y) and (x, -y) are present on the same graph. A graph will have x-axis symmetry if we get an equivalent equation to the original one when we replace y with –y.
Let us test the symmetry of a graph with the equation 6 + xy4 = 0
x-axis symmetry test: replacing y with –y:
6 + x(-y)4 = 0
⇒ 6 + xy4 = 0 (this is the original equation)
Thus the graph is symmetric about the x-axis.
y-axis Symmetry
A graph is symmetric about the y-axis when the points (x, y) and (-x, y) are present on the same graph. A graph will have y-axis symmetry if we get an original equation to the original one when we replace x with –x.
Let us test the symmetry of a graph with the equation y = x4 +3x2
x-axis symmetry test: replacing y with –y:
-y = x4 +3x2
Y = – (x4 +3x2), this is not the equivalent equation.
y-axis symmetry test: replacing x with –x.
So, y = (-x)4 +3(-x)2
y = x4 +3x2 (this is the original equation)
Thus the graph is symmetric about the y-axis.
Origin Symmetry
A graph is symmetric about the origin when the points (x, y) and (-x,-y) are present on the same graph. A graph will have symmetry about the origin if we get an equivalent equation to the original one when we replace y with –y and also x with –x.
Let us test the symmetry of a graph with the equation xy = 4
x-axis symmetry test: replacing y with –y:
x(-y) = 4
-xy = 4, (this is not the original equation)
y-axis symmetry test: replacing x with –x.
(-x)y = 4
-xy = 4, (this is also not the original equation)
Origin symmetry test: replacing x with –x and y with -y.
(-x)(-y) = 4
xy = 4, (this is the original equation)
So, the graph is symmetric about the origin.
How to Find the Axis of Symmetry on a Graph
For a parabola the axis of symmetry always passes through the vertex. The x -coordinate of the vertex is the equation of the axis of symmetry of the parabola.
For a quadratic function in standard form y = ax2 + bx + c, the axis of symmetry is x = -b/2a
Now, let us find the axis of symmetry of the given quadratic graph.
How to Find Axis of Symmetry on a Graph
The given graph shows line symmetry about the y-axis.
So for a graph y = x2 – 4
Axis of symmetry is x = -b/2a, here a = 1, b = 0
∴ x = 0
Solved Examples
Prove the symmetry of the graph y = x2
Solution:
x-axis symmetry test: Replacing y with –y, -y = x2 (this is not equivalent to the original equation) y-axis symmetry test: Replacing x with –x, y = (-x) 2 = x2 ∴ The graph is symmetric about the y-axis
Test the symmetry of the graph y = x3
Solution:
x-axis symmetry test: Replacing y with –y, -y = x3 (this is not equivalent to the original equation) (this is not equivalent to the original equation) y = (-x)3 = -x3 (this is not equivalent to the original equation) Origin symmetry test: replacing x with –x and y with -y. -y = (-x)3 -y = -x3 Y = x3, (this is the original equation) ∴ The graph is symmetric about the origin.
Which graph shows rotational symmetry and which one shows line symmetry in the figure alongside?
Solution:
Graph 2 shows rotational symmetry since the graph is a perfect circle about the origin. Graph 1 shows point symmetry where the central point is the origin.
Which graph shows line symmetry about the y-axis in the figure alongside?
Solution:
Graph 2 shows line symmetry about the y-axis. Graph 1 does not have the y-axis as its line of symmetry.
Which graph shows rotational symmetry and which one shows line symmetry in the figure alongside?
Which graph shows line symmetry about the y-axis in the figure alongside?