how can you tell if the function is symmetric with respects to the x and y axis, and the origin?

To determine if a function is symmetric with respect to the x-axis, we need to check if it is unchanged when reflected over the x-axis. Similarly, to determine if a function is symmetric with respect to the y-axis, we need to check if it remains the same when reflected over the y-axis. Lastly, to determine if a function is symmetric with respect to the origin, we need to check if it remains invariant when reflected over the origin.

To check for symmetry with respect to the x-axis, we can follow these steps:

1. Replace every occurrence of y in the equation with -y.
2. Simplify the equation if possible.
3. If the resulting equation is equivalent to the original equation, the function is symmetric with respect to the x-axis.

For example, let's consider the function y = x^2:

1. Replace y with -y: -y = x^2.
2. Simplify: y = -x^2.
3. Since the resulting equation is not equivalent to the original equation, the function y = x^2 is not symmetric with respect to the x-axis.

To check for symmetry with respect to the y-axis, follow these steps:

1. Replace every occurrence of x in the equation with -x.
2. Simplify the equation if possible.
3. If the resulting equation is equivalent to the original equation, the function is symmetric with respect to the y-axis.

For example, let's consider the function y = x^3:

1. Replace x with -x: y = (-x)^3.
2. Simplify: y = -x^3.
3. Since the resulting equation is not equivalent to the original equation, the function y = x^3 is not symmetric with respect to the y-axis.

To check for symmetry with respect to the origin, we need to satisfy both the x-axis and y-axis symmetry conditions. This means that the function needs to be symmetric with respect to both the x-axis and the y-axis simultaneously.

For example, let's consider the function y = x^4:

1. Replace y with -y: -y = x^4.
2. Replace x with -x in the resulting equation: -y = (-x)^4.
3. Simplify: -y = x^4.
4. Replace y with -y again: -(-y) = x^4.
5. Simplify: y = x^4.
6. Since the resulting equation is equivalent to the original equation, the function y = x^4 is symmetric with respect to the origin.

By following these steps, you can determine if a function is symmetric with respect to the x-axis, y-axis, or the origin.