Axis Symmetry Notes

Y-axis symmetry

• Definition: The graph is symmetric about the y-axis if for every point (x, y) on the graph, the point (-x, y) is also on the graph. For a function y = f(x), this is equivalent to $f(-x) = f(x)$ for all x in the domain.

• How to test (with respect to the y-axis): Replace x with -x in the equation. If after simplification the equation is the same as the original, the graph is symmetric about the y-axis.

• Key takeaway: Y-axis symmetry corresponds to even functions.

• Example 1: y = x^2

  • Original: $y = x^2$

  • Test: $f(-x) = (-x)^2 = x^2 = f(x)$

  • Conclusion: symmetric about the y-axis.

• Example 2: y = x^4

  • Test: $f(-x) = (-x)^4 = x^4 = f(x)$

  • Conclusion: symmetric about the y-axis.

• Non-example: y = x^3

  • Test: f(−x)=(−x)3=−x3f(−x)=(−x)3=−x3 This is not equal to f(x)f(x).

  • Conclusion: not symmetric about the y-axis.

• Important nuance about signs:

  • With negative x, raising to an even power turns negative into positive; with odd power it stays negative; This helps explain symmetry.

• Parens note:

  • The expression (-x)^3 denotes cubing the negative quantity, which yields -(x^3).

  • It is distinct from writing -x^3 without parentheses, since the latter is interpreted as -(x^3) as well, but parentheses help clarify the scope of the exponent.

X-axis symmetry

• Definition: The graph is symmetric about the x-axis if for every point (x, y) on the graph, the point (x, -y) is also on the graph. For the function y = f(x), this translates to the transformed equation after replacing y with -y being equivalent to the original; in practice, this is rarely satisfied for nontrivial functions.

• Test: Replace y with -y in the original equation and simplify. If the resulting equation is algebraically equivalent to the original, x-axis symmetry exists.

• Example 1: y = x^2

  • Original: $y = x^2$

  • After substitution: $$-y = x^2 \