Recognize symmetry in coordinate graphs

Symmetry in a coordinate graph means a figure or set of plotted integer points remains unchanged when reflected across a coordinate axis, with corresponding points sharing one coordinate and having opposite values for the other; simple symmetry about the origin reverses both coordinate signs. This distinguishes actual coordinate relationships from visual balance and supports later work with transformations, without extending to arbitrary lines of symmetry, formal transformation rules, or algebraic proofs.

Detailed Explanation: Recognize symmetry in coordinate graphs

Symmetry means a figure looks unchanged after it is reflected across an axis or the origin. To check symmetry, compare the coordinates of the plotted points—not just how the picture looks.

Across the $x$ -axis: the $x$ -coordinate stays the same, and the $y$ -coordinate changes sign.

(x,y)\longleftrightarrow(x,-y)

Across the $y$ -axis: the $y$ -coordinate stays the same, and the $x$ -coordinate changes sign.

(x,y)\longleftrightarrow(-x,y)

About the origin: both coordinate signs change.

(x,y)\longleftrightarrow(-x,-y)

Worked example

A graph contains these points:

(-3,2),\quad (3,2),\quad (-3,-2),\quad (3,-2)

Step 1: Check the $y$ -axis.

For $((-3,2))$ , the matching point across the $y$ -axis should be $((3,2))$ . It is on the graph.

For $((-3,-2))$ , the matching point should be $((3,-2))$ . It is also on the graph.

So the points are symmetric across the $y$ -axis.

Step 2: Check the $x$ -axis.

For $((-3,2))$ , the matching point across the $x$ -axis should be $((-3,-2))$ . It is on the graph.

For $((3,2))$ , the matching point should be $((3,-2))$ . It is also on the graph.

So the points are symmetric across the $x$ -axis.

Step 3: Check the origin.

The point opposite $((-3,2))$ through the origin is $((3,-2))$ . It is on the graph.

The point opposite $((3,2))$ is $((-3,-2))$ . It is also on the graph.

Therefore, this set of points is symmetric across the $x$ -axis, symmetric across the $y$ -axis, and symmetric about the origin.

Learn by doing: Recognize symmetry in coordinate graphs

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Cartesian Grid - Reflection of Point (Grid to Grid) across Axis

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