Without graphing, 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. See Examples 3 and 4. y=x^2+5

Verified step by step guidance

1. Identify the type of symmetry to test: x-axis, y-axis, and origin.
2. For x-axis symmetry, replace y with -y in the equation and check if the equation remains unchanged. Substitute -y for y: -y = x^2 + 5. Simplify to see if it matches the original equation.
3. For y-axis symmetry, replace x with -x in the equation and check if the equation remains unchanged. Substitute -x for x: y = (-x)^2 + 5. Simplify to see if it matches the original equation.
4. For origin symmetry, replace both x with -x and y with -y in the equation and check if the equation remains unchanged. Substitute -x for x and -y for y: -y = (-x)^2 + 5. Simplify to see if it matches the original equation.
5. Determine which, if any, of the symmetries apply based on the results of the substitutions and simplifications.

Key Concepts

Symmetry with respect to the x-axis

An equation is symmetric with respect to the x-axis if replacing y with -y results in an equivalent equation. This means that for every point (x, y) on the graph, the point (x, -y) will also be on the graph. This type of symmetry indicates that the graph will reflect across the x-axis.

Symmetry with respect to the y-axis

An equation is symmetric with respect to the y-axis if replacing x with -x yields an equivalent equation. This implies that for every point (x, y) on the graph, the point (-x, y) will also be present. Such symmetry suggests that the graph will reflect across the y-axis.

Symmetry with respect to the origin

An equation is symmetric with respect to the origin if replacing both x with -x and y with -y results in an equivalent equation. This means that for every point (x, y) on the graph, the point (-x, -y) will also be on the graph. This type of symmetry indicates that the graph will reflect through the origin.