lesson
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If you fold a butterfly's wings, they line up perfectly, and if you spin a playing card halfway around, it looks identical. Graphs of functions can have these exact same superpowers: line symmetry and point symmetry.
How can you tell just by looking at a curve whether it reflects like a mirror or rotates like a propeller?
Line Symmetry Across a Vertical Line
A graph has line symmetry (or reflectional symmetry) across a line x=h if folding the graph along that vertical line makes both halves match up completely. When that mirror line is the y-axis (x=0), the function is called an even function.
This mirror reflection means moving right to x gives the exact same height as moving left to −x, which gives the algebraic rule f(−x)=f(x) for all inputs in the domain.
📊Interactive/animated coordinate plane showing the even function graph y = x^2 - 2. A dashed vertical line marks the y-axis (x = 0) with a mirror icon at the top. Two points are plotted on the curve: (2, 2) on the right and (-2, 2) on the left. A light blue horizontal dashed line connects them across the y-axis, highlighting equal distances from the center. Text labels show (x, y) and (-x, y) with a folding animation cue.
Let's test f(x)=x4−2x2+1. Because f(−x)=(−x)4−2(−x)2+1=x4−2x2+1=f(x), every point (a,b) on the right side has an identical twin (−a,b) on the left side, proving mirror symmetry across the y-axis.
What happens if a graph doesn't fold onto itself across a line, but instead spins onto itself when turned upside down?
Point Symmetry About the Origin
A graph has point symmetry (or 180∘ rotational symmetry) around a point (h,k) if rotating the entire graph halfway around that point leaves it unchanged. When that center of rotation is the origin (0,0), the function is called an odd function.
Because a half-turn flips both horizontal and vertical directions, moving right to x gives the opposite height of moving left to −x, which leads to the algebraic rule f(−x)=−f(x).
📊Coordinate plane showing the cubic function y = x^3 - 3x. The origin (0, 0) is highlighted with a central rotation point and a circular arrow indicating a 180-degree spin. A point (2, 2) in the first quadrant is connected through the origin by a straight dashed line to its rotational pair (-2, -2) in the third quadrant. Labels clearly show (x, y) mapping to (-x, -y) through the origin.
Let's test f(x)=x3−3x. Substituting −x gives f(−x)=(−x)3−3(−x)=−x3+3x=−(x3−3x)=−f(x), confirming that spinning the graph 180∘ around (0,0) lands every curve back onto itself.