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. 5y^2 + 5x^2 =30

Verified step by step guidance

1. Identify the given equation: $5y^2 + 5x^2 = 30$.
2. Divide the entire equation by 30 to simplify: $\frac{5y^2}{30} + \frac{5x^2}{30} = 1$.
3. Simplify the equation: $\frac{y^2}{6} + \frac{x^2}{6} = 1$.
4. Recognize the equation as an ellipse centered at the origin with equal semi-major and semi-minor axes, indicating symmetry about both the x-axis and y-axis.
5. Conclude that the graph is symmetric with respect to the x-axis, y-axis, and 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 replacing both x and y with their negatives results in the same equation.

Identifying Symmetry

To determine the type of symmetry a graph possesses, one can apply specific tests. For x-axis symmetry, substitute -y for y; for y-axis symmetry, substitute -x for x; and for origin symmetry, substitute both -x and -y. If the resulting equation is equivalent to the original, the graph exhibits that type of symmetry.

Conic Sections

The given equation, 5y^2 + 5x^2 = 30, represents a conic section, specifically a circle when rearranged. Understanding the standard forms of conic sections helps in identifying their properties, including symmetry. A circle is symmetric with respect to both axes and the origin, which is crucial for analyzing the graph's symmetry.