Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. $5 y^{2} + 5 x^{2} = 30$

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Rewrite the given equation for clarity: $5y^2 + 5x^2 = 30$.

To check symmetry with respect to the y-axis, replace $x$ with $-x$ and see if the equation remains unchanged: $5y^2 + 5(-x)^2 = 30$.

To check symmetry with respect to the x-axis, replace $y$ with $-y$ and see if the equation remains unchanged: $5(-y)^2 + 5x^2 = 30$.

To check symmetry with respect to the origin, replace both $x$ with $-x$ and $y$ with $-y$ and see if the equation remains unchanged: $5(-y)^2 + 5(-x)^2 = 30$.

Compare the resulting equations from each substitution to the original equation to determine which symmetries hold true.

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주요 개념

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Symmetry with Respect to the x-axis

A graph is symmetric about the x-axis if replacing y with -y in the equation yields an equivalent equation. This means the graph looks the same above and below the x-axis, reflecting points across it.

Symmetry with Respect to the y-axis

A graph is symmetric about the y-axis if replacing x with -x in the equation results in the same equation. This indicates the graph mirrors itself on the left and right sides of the y-axis.

Symmetry with Respect to the Origin

A graph is symmetric about the origin if replacing both x with -x and y with -y produces the original equation. This means the graph is unchanged when rotated 180 degrees around the origin.

Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. 5y2+5x2=305y^2 + 5x^2 =30