The symmetry of a linear graph depends on the specific line. A linear graph can exhibit symmetry with respect to a point or another line.
Symmetry Explained
A graph is symmetric with respect to a line if reflecting the graph over that line results in the same graph. This line is referred to as the axis of symmetry. Here's a breakdown:
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Symmetry with Respect to a Point: A linear graph (a straight line) can have point symmetry. This occurs if the line passes through the origin (0, 0). In this case, reflecting the line through the origin leaves it unchanged. Therefore, the origin serves as the point of symmetry.
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Symmetry with Respect to a Line: A linear graph has symmetry with respect to itself. This is somewhat trivial, but it's technically accurate. Also, certain linear graphs, such as horizontal lines (y = c) or vertical lines (x = c), exhibit symmetry across a line. For a horizontal line, any vertical line, x = a, will serve as an axis of symmetry. For a vertical line, any horizontal line, y = a, will serve as an axis of symmetry.
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Symmetry with Respect to the Y-Axis: Only the horizontal line y=0 (the x-axis) is symmetric with respect to the y-axis.
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Symmetry with Respect to the X-Axis: Only the vertical line x=0 (the y-axis) is symmetric with respect to the x-axis.
Examples
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y = x: This line is symmetric about the origin.
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y = 5: This horizontal line is symmetric about any vertical line. For example, x = 0 (the y-axis), x = 1, x = -2, etc.
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x = 3: This vertical line is symmetric about any horizontal line. For example, y = 0 (the x-axis), y = 1, y = -2, etc.
In summary, while not all linear graphs have line symmetry (besides symmetry with respect to themselves), a linear graph can exhibit symmetry depending on its equation and the point or line being considered. Generally, the specific symmetry exhibited depends on the orientation of the line.
