Determine whether the graph of each equation is symmetric with respect to the y-axis, the x-axis, the origin, more than one of those, or none of these. x^2 + y^2 =17

Verified step by step guidance

1. To determine symmetry with respect to the y-axis, replace x with -x in the equation. Substitute -x for x in the equation x^2 + y^2 = 17, resulting in (-x)^2 + y^2 = 17. Simplify the expression to check if it matches the original equation.
2. To determine symmetry with respect to the x-axis, replace y with -y in the equation. Substitute -y for y in the equation x^2 + y^2 = 17, resulting in x^2 + (-y)^2 = 17. Simplify the expression to check if it matches the original equation.
3. To determine symmetry with respect to the origin, replace both x with -x and y with -y in the equation. Substitute -x for x and -y for y in the equation x^2 + y^2 = 17, resulting in (-x)^2 + (-y)^2 = 17. Simplify the expression to check if it matches the original equation.
4. After performing the substitutions and simplifications for each type of symmetry, compare the resulting equations to the original equation x^2 + y^2 = 17. If the resulting equation matches the original, the graph is symmetric with respect to that axis or the origin.
5. Based on the results of the comparisons, conclude whether the graph is symmetric with respect to the y-axis, the x-axis, the origin, more than one of these, or none of these.

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 y-axis if replacing x with -x in the equation yields the same equation. Similarly, it is symmetric with respect to the x-axis if replacing y with -y results in the same equation, and it is symmetric with respect to the origin if replacing both x and y with their negatives preserves the equation.

Circle Equation

The equation x^2 + y^2 = r^2 represents a circle centered at the origin with radius r. In this case, the equation x^2 + y^2 = 17 describes a circle with a radius of √17. Understanding the standard form of a circle's equation is crucial for analyzing its symmetry properties, as circles inherently exhibit symmetry about both axes and the origin.

Testing for Symmetry

To determine the symmetry of a graph, specific tests can be applied. For y-axis symmetry, substitute -x into the equation; for x-axis symmetry, substitute -y; and for origin symmetry, substitute both -x and -y. If the resulting equation remains unchanged for any of these substitutions, the graph exhibits that type of symmetry. This methodical approach is essential for analyzing the given equation.