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. Check symmetry with respect to the x-axis by replacing y with -y in the equation and see if the equation remains unchanged. The equation becomes (-y)^3 = x + 4, which simplifies to -y^3 = x + 4. This is not the same as the original equation, so the graph is not symmetric with respect to the x-axis.
2. Check symmetry with respect to the y-axis by replacing x with -x in the equation and see if the equation remains unchanged. The equation becomes y^3 = -x + 4, which is not the same as the original equation, so the graph is not symmetric with respect to the y-axis.
3. Check symmetry with respect to the origin by replacing x with -x and y with -y in the equation and see if the equation remains unchanged. The equation becomes (-y)^3 = -x + 4, which simplifies to -y^3 = -x + 4. This is not the same as the original equation, so the graph is not symmetric with respect to the origin.
4. Since the equation does not remain unchanged under any of the symmetry operations (replacing y with -y, x with -x, or both), the graph of the equation y^3 = x + 4 is not symmetric with respect to the x-axis, y-axis, or the origin.
5. Conclude that the graph of the equation y^3 = x + 4 has no symmetry with respect to the x-axis, y-axis, or the origin.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. For example, a graph is symmetric with respect to the x-axis if replacing y with -y yields the same equation, and symmetric with respect to the y-axis if replacing x with -x does. Origin symmetry occurs if both transformations hold true simultaneously.

Testing for Symmetry

To test for symmetry, one can substitute values into the equation. For x-axis symmetry, replace y with -y and check if the equation remains valid. For y-axis symmetry, replace x with -x, and for origin symmetry, replace both x and y with their negatives. The results of these substitutions determine the type of symmetry present in the graph.

Understanding Implicit Functions

The equation y^3 = x + 4 is an implicit function, meaning it defines y in terms of x without explicitly solving for y. Analyzing implicit functions often requires algebraic manipulation to explore their properties, including symmetry. Understanding how to work with implicit equations is crucial for determining their graphical characteristics.