One way to represent how an isometry (or any other transformation of space) transforms the plane (or any space) is through analytic geometry:[br][br](A) Introduce a coordinate system in the plane[br](B) Identify points, P, in the plane by their coordinates: P(x,y)[br](C) Specify the coordinates for the new point, P', to which P is mapped by the isometry: P'(x',y').[br][br]So the isometry is represented by the mapping P(x,y)-->P'(x',y'). [br][br]Use the applet below to determine where the point, P(x,y), is mapped by the following isometries:[br][br]1) Reflection over any of the lines of reflection shown in the applet;[br]2) Rotation about the origin by 90 degrees or 180 degrees;[br]3) Translation in any direction.

Now, what if we tried to represent a more complicated transformation, like a reflection over the line y=x+4? Where would the point P(x,y) go then? One idea we might try is to reduce this transformation into the prior transformation by, first, translating the line of reflection down 4 so that it passes through the origin. But then we would need to undo this transformation after performing the reflection. This is called conjugation: a[sup]-1[/sup]ba, where a and b are transformations and a[sup]-1[/sup] is the inverse transformation of a.[br][br]Does this work in the present case? How can you check?[br][br]Would this conjugation strategy always work? Can you justify it?