Symmetry Determine whether the graphs of the following...

Symmetry Determine whether the graphs of the following equations and functions have symmetry about the $x$-axis, the $y$ -axis, or the origin. Check your work by graphing.
$$x^{2 / 3}+y^{2 / 3}=1$$

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Key Concepts

Even and Odd Functions

Even and odd functions relate closely to symmetry. Even functions, which satisfy f(-x) = f(x), exhibit symmetry about the y-axis, while odd functions, which satisfy f(-x) = -f(x), have symmetry about the origin. Although these definitions specifically apply to functions expressed in the form y = f(x), they provide intuition on the type of symmetry a graph might display.

Graph Symmetry

Graph symmetry refers to the property of a graph remaining unchanged under certain geometric transformations, such as reflections across an axis or rotation about a point. Recognizing symmetry can simplify analysis and graphing, as only a part of the graph may need to be drawn with the remainder inferred by the symmetry.

Reflection Tests

Reflection tests are methods used to determine symmetry. To check for y-axis symmetry, every x in the equation is replaced by -x; for x-axis symmetry, every y is replaced by -y; and for origin symmetry, both x and y are replaced by -x and -y respectively. If the equation remains unchanged after a substitution, the graph has the corresponding symmetry.

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