2.6
Symmetry
The equation of an ellipse centered at the origin represents a shape in which every point on the curve is at a constant sum of distances from two points called foci, and every point has a symmetric match across the axes.
Like an elliptical planetary orbit, this ellipse forms a smooth curve stretching farther along the x-axis than the y-axis, giving it a horizontal orientation.
The graph displays symmetry in three distinct ways: across the x-axis, across the y-axis, and about the origin.
Symmetry across the x-axis appears when the graph reflects top to bottom, forming equal halves above and below the horizontal axis.
The equation remains unchanged when y is replaced with its negative, confirming this symmetry.
Symmetry across the y-axis reflects the shape from left to right, maintaining balance across the vertical axis.
This is confirmed when x is replaced with its negative, and the equation remains the same.
Symmetry about the origin, called C2 rotational symmetry, occurs when a one-hundred-eighty-degree rotation leaves the graph unchanged but rotated.
When both x and y are replaced with their negatives, the equation still holds, confirming this symmetry.
The equation of an ellipse centered at the origin defines all points whose distances from the center maintain a constant ratio between the horizontal and vertical axes. This equation results in a smooth, closed curve that extends further along the x-axis than the y-axis, giving it a horizontal orientation. Such an ellipse demonstrates three kinds of symmetry: across the x-axis, across the y-axis, and about the origin. These symmetries are essential in understanding the graph's structure and behavior.
Symmetry across the x-axis is observed when the graph reflects over the horizontal axis, forming identical upper and lower halves. Algebraically, this is confirmed when replacing y with −y in the ellipse's equation yields an equivalent expression. Similarly, symmetry across the y-axis is evident when the graph reflects over the vertical axis, producing equal left and right halves; this is validated when replacing x with −x does not change the equation. Furthermore, symmetry about the origin indicates that the graph remains unchanged in form after a 180° rotation. This rotational symmetry is demonstrated when both x and y are negated simultaneously, and the equation still holds.
These types of symmetry reflect the ellipse's underlying geometric consistency and allow for efficient graphing and analysis. The unchanged equation under these transformations confirms the ellipse's balanced and predictable structure, reinforcing its mathematical elegance.
The equation of an ellipse centered at the origin represents a shape in which every point on the curve is at a constant sum of distances from two points called foci, and every point has a symmetric match across the axes.
Like an elliptical planetary orbit, this ellipse forms a smooth curve stretching farther along the x-axis than the y-axis, giving it a horizontal orientation.
The graph displays symmetry in three distinct ways: across the x-axis, across the y-axis, and about the origin.
Symmetry across the x-axis appears when the graph reflects top to bottom, forming equal halves above and below the horizontal axis.
The equation remains unchanged when y is replaced with its negative, confirming this symmetry.
Symmetry across the y-axis reflects the shape from left to right, maintaining balance across the vertical axis.
This is confirmed when x is replaced with its negative, and the equation remains the same.
Symmetry about the origin, called C2 rotational symmetry, occurs when a one-hundred-eighty-degree rotation leaves the graph unchanged but rotated.
When both x and y are replaced with their negatives, the equation still holds, confirming this symmetry.
From Chapter 2:
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