Understanding Symmetry: Graph and Equation Symmetry with the Origin

symmetric with origin

When a graph or equation is said to be symmetric with respect to the origin, it means that the graph or equation exhibits symmetry when reflected across the origin (0,0) on a coordinate plane

When a graph or equation is said to be symmetric with respect to the origin, it means that the graph or equation exhibits symmetry when reflected across the origin (0,0) on a coordinate plane.

To determine if a graph or equation is symmetric with respect to the origin, we can follow these steps:

1. Graphical Method:
– Plot a few points on each side of the origin.
– Reflect these points across the origin by negating their x and y coordinates.
– If the reflected points lie on the graph, then the graph is symmetric with the origin.
– Alternatively, if the graph remains unchanged after reflecting it across the origin, it is symmetric with respect to the origin.

2. Algebraic Method:
– For an equation in standard form, f(x,y) = 0, substitute (-x,-y) for (x,y) in the equation.
– Simplify the equation.
– If the equation remains the same, then the equation is symmetric with respect to the origin.
– For example, if the equation is x^2 + y^2 = 1, replacing x with -x and y with -y gives (-x)^2 + (-y)^2 = 1, which simplifies to x^2 + y^2 = 1, indicating symmetry with the origin.

It’s important to note that a graph or equation can exhibit other types of symmetry as well, such as symmetry with respect to the x-axis or the y-axis. To determine the type of symmetry, it’s necessary to examine the behavior of the graph or equation in relation to these axes.

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