standard transformations as follows: Changing to moves the graph to the right units. Changing to moves the graph up units. Changing to stretches the graph horizontally by a factor of . (think of the as being dilated) Changing to stretches the graph vertically. Changing to and changing to rotates the graph by an angle . There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations. For example, the parent function has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if , then it can be transformed into . In the new transformed function, is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative values, the function is reflected in the -axis. The value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like , reflects the function in the -axis when it is negative. The and values introduce translations, , vertical, and horizontal. Positive and values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end. Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations. Suppose that is a relation in the plane. For example, is the relation that describes the unit circle. Finding intersections of geometric objects For two geometric objects P and Q represented by the relations and the intersection is the collection of all points which are in both relations. For example, might be the circle with radius 1 and center : and might be the circle with radius 1 and center . The intersection of these two circles is the collection of points which make both equations true. Does the point make both equations true? Using for , the equation for becomes or which is true, so is in the relation . On