Could somebody please check this for me?
Dtermine whether the graph of each equation is symmetric with respect to the origin, the x-axis, the y-axis, and the line y=x or the line y=-x.
I came up with the line being symmetric to the x-axis, the y-axis, and the line y=x. I'm not completely sure if I did the others right.
We can't help you with this question unless you provide the equations that are being evaluated for symmetry.
Oh, I'm so sorry; thanks for your time. The equation was x^2+y^2=9
To determine whether a graph is symmetric with respect to the origin, the x-axis, the y-axis, the line y=x, or the line y=-x, you can follow these steps:
1. Origin Symmetry: Substitute (-x, -y) into the equation and simplify. If the resulting equation is equivalent to the original equation, then the graph is symmetric with respect to the origin.
2. X-Axis Symmetry: Replace y with (-y) in the equation and simplify. If the resulting equation is equivalent to the original equation, then the graph is symmetric with respect to the x-axis.
3. Y-Axis Symmetry: Replace x with (-x) in the equation and simplify. If the resulting equation is equivalent to the original equation, then the graph is symmetric with respect to the y-axis.
4. Line y=x Symmetry: Replace x with y and y with x in the equation. Simplify the resulting equation. If it is equivalent to the original equation, then the graph is symmetric with respect to the line y=x.
5. Line y=-x Symmetry: Replace x with (-y) and y with (-x) in the equation. Simplify the resulting equation. If it is equivalent to the original equation, then the graph is symmetric with respect to the line y=-x.
Now, let's assess your results. You mentioned that you believe the graph is symmetric with respect to the x-axis, y-axis, and line y=x. To verify this, plug the appropriate values into the equation and compare.
For example, if the equation is y = x^2, let's evaluate the symmetry:
1. Origin Symmetry: (-x)^2 = x^2. Both equations are equivalent, so it is symmetric with respect to the origin.
2. X-Axis Symmetry: (-y) = x^2. This equation is not equivalent to the original, so the graph is not symmetric with respect to the x-axis.
3. Y-Axis Symmetry: y = (-x)^2. Both equations are equivalent, so it is symmetric with respect to the y-axis.
4. Line y=x Symmetry: x = y^2. This equation is not equivalent to the original, so the graph is not symmetric with respect to the line y=x.
5. Line y=-x Symmetry: x = (-y)^2. Both equations are equivalent, so it is symmetric with respect to the line y=-x.
Based on these evaluations, your conclusion is partially correct. The graph is symmetric with respect to the origin, the y-axis, and the line y=-x. However, it is not symmetric with respect to the x-axis or the line y=x.