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Seeing Symmetry

For Odd Functions, there are two ways to see the symmetry with respect to the origin

In the figure above, there are two ways to look at the symmetry of the odd function with respect to the origin. One way is that for every point (x, f(x)), there is a point (-x, -f(x)) on the opposite side of the origin (so f(-x) = -f(x)). A second way is to think of rotating the graph an angle of 180 degrees about the origin. If the second graph is exactly the same as the first graph, there is symmetry about the origin.

What about changing direction?

For a vector starting at the origin, that is just going in 1 direction, rotating 180 degrees will have the vector go in the exact opposite direction. When the Bible speaks of repentance of sin, it is not merely saying "I am sorry for the sin committed." It is a pledge to completely change direction and go in the opposite direction, now doing what God wants us to do rather than what we, as sinful people, want to do. Acts 20:21: I have declared to both Jews and Greeks that they must turn to God in repentance and have faith in our Lord Jesus.