Question

How can you algebraically show that a graph has x-axis, y-axis, or origin symmetry?

Answer 1

To algebraically show symmetry in a graph, for x-axis symmetry substitute -x for x in the equation, for y-axis symmetry substitute -y for y, and for origin symmetry substitute -x for x and -y for y. Check if the resulting equations are equivalent to the original equation.

Step-by-step explanation:

To algebraically show that a graph has x-axis symmetry, you can substitute -x for x in the equation and check if the resulting equation is equivalent to the original equation. If it is, then the graph is symmetric with respect to the x-axis. For example, if the equation is y = x^2, substituting -x for x gives y = (-x)^2 = x^2, which is equivalent to the original equation. Therefore, the graph has x-axis symmetry.

To algebraically show that a graph has y-axis symmetry, you can substitute -y for y in the equation and check if the resulting equation is equivalent to the original equation. If it is, then the graph is symmetric with respect to the y-axis. For example, if the equation is y = x^2, substituting -y for y gives -y = x^2, which is not equivalent to the original equation. Therefore, the graph does not have y-axis symmetry.

To algebraically show that a graph has origin symmetry, you can substitute -x for x and -y for y in the equation and check if the resulting equation is equivalent to the original equation. If it is, then the graph is symmetric with respect to the origin. For example, if the equation is y = x^2, substituting -x for x and -y for y gives -y = (-x)^2, which is equivalent to the original equation. Therefore, the graph has origin symmetry.

Answer 2

To algebraically determine the symmetry of a graph, examine the functions f(x), f(-x), f(-y), and f(y).

If f(x) = f(-x), the graph has even symmetry about the y-axis.

If f(-x) = -f(x), the graph has odd symmetry about the origin.

If f(y) = f(-y), the graph has even symmetry about the x-axis.

Step-by-step explanation:

To determine symmetry algebraically, consider the given function f(x) and assess its behavior when manipulated with respect to x and y values. If f(x) = f(-x), this demonstrates even symmetry about the y-axis. For instance, in the function f(x) = x² , substituting -x for x yields f(-x) = (-x)² = x² , proving symmetry about the y-axis.

Alternatively, for odd symmetry about the origin, check if f(-x) = -f(x). A function such as f(x) = x³ satisfies this condition: f(-x) = (-x)³ = -x³ = -f(x), showcasing symmetry about the origin.

Lastly, assess even symmetry about the x-axis by examining f(y) = f(-y). In the function f(x) = cos(y), f(y) = cos(y) and f(-y) = cos(-y) = cos(y), illustrating symmetry about the x-axis as both sides are equal.

In summary, analyzing how the function behaves when replacing x with -x or y with -y helps ascertain the type of symmetry (about the y-axis, origin, or x-axis) present in a graph. These algebraic manipulations provide a clear understanding of the symmetry exhibited by the graph without needing to graphically represent it.