Graphing techniques

Check the graph and its symmetries · Diagrams

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Reflection axes: circle and cubic

Circle and cubic symmetry

Choose a curve and a mirror line, then move point P.

Curve

Mirror line

Line angle through origin0°

Point angle45°

P = (3.53553, 3.53553)P′ = (3.53553, -3.53553)The reflected point lies on the curve.

A matching point is not a proof of symmetry.

Why this works

Every straight line through the origin is a reflection axis. Equal axis scales keep this circle round.

A matching point alone does not prove reflection symmetry. Compare the whole curve; the overlay is clipped to the viewing window. Solid and dashed lines identify the original and reflected curves.

Circle x²+y²=25 with intercepts ±5 is symmetric in every line through the origin; horizontal and vertical examples shown. Cubic y=x³+x² has no reflection axis; intercept −1 and origin, with corrected exact maximum (−2/3,4/27). Printed −0.677 is explicitly recorded as a source slip.

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