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.
$y^3=x+4$

Verified step by step guidance

1. Rewrite the given equation: \(y^3 = x + 4\).
2. To test symmetry about the y-axis, replace \(x\) with \(-x\) and check if the equation remains unchanged: \(y^3 = -x + 4\).
3. 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 \(-y^3 = x + 4\).
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 \(-y^3 = -x + 4\).
5. Compare the transformed equations to the original to determine which symmetry (x-axis, y-axis, origin) the graph has, or if it has none.

Key Concepts

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.