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.
$y^{3} = x + 4$

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Rewrite the given equation: $y^3 = x + 4$.

To test symmetry about the y-axis, replace $x$ with $-x$ and check if the equation remains unchanged: $y^3 = -x + 4$.

To test symmetry about the x-axis, replace $y$ with $-y$ and check if the equation remains unchanged: $(-y)^3 = x + 4$, which simplifies to $-y^3 = x + 4$.

To test symmetry about the origin, replace $x$ with $-x$ and $y$ with $-y$ simultaneously and check if the equation remains unchanged: $(-y)^3 = -x + 4$, which simplifies to $-y^3 = -x + 4$.

Compare the transformed equations to the original to determine which symmetry (x-axis, y-axis, origin) the graph has, or if it has none.

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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 that for every point (x, y) on the graph, the point (x, -y) is also on the graph, reflecting the graph across the x-axis.

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 implies that for every point (x, y), the point (-x, y) is also on the graph, reflecting the graph across 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 in the equation produces an equivalent equation. This means that for every point (x, y), the point (-x, -y) is also on the graph, indicating rotational symmetry of 180 degrees about 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.y3=x+4y^3 = x + 4