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**Checking an equation for symmetry**

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**Types of symmetry Three types of symmetry**

Symmetry with respect to the y-axis (vertical) Symmetry with respect to the x-axis (horizontal) Symmetry with respect to the origin

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**Checking for symmetry with respect to the y-axis**

In the original equation, replace x with –x, and simplify. If the result is the original equation, there is symmetry about the y-axis. Example y = x2 – 2 y = (-x)2 – 2 Substitute –x for x y = x2 – 2 Simplify There is symmetry about the y-axis.

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**Checking for symmetry with respect to the y-axis**

Try this one. y = 3 – 2x y = 3 - 2(-x) y = 3 + 2x There is not symmetry about the y-axis.

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**Check for symmetry about the x-axis**

In the original equation, replace y with –y, solve for y. If the new equation is the same, there is symmetry about the x-axis. y2 = x + 4 (-y)2 = x + 4 There is symmetry about the x-axis.

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**Check for symmetry about the x-axis**

Try this one. y = x3 – 2x (-y) = x3 – 2x y = -x3 + 2x There is not symmetry about the x-axis.

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**Checking for symmetry about the origin**

In the original equation, substitute –x for x and –y for y, then simplify the equation. If the new equation is the same as the original, there is symmetry about the origin. y = x3 – 4x -y = (-x)3 – 4(-x) -y = -x3 + 4x There is symmetry about the origin.

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**Checking for symmetry about the origin**

Try this one. y2 = x – 1 (-y)2 = (-x) – 1 y2 = -x – 1 There is no symmetry about the origin.

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Symmetry Two points, P and P ₁, are symmetric with respect to line l when they are the same distance from l, measured along a perpendicular line to l.

Symmetry Two points, P and P ₁, are symmetric with respect to line l when they are the same distance from l, measured along a perpendicular line to l.

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