Editing Analytic geometry (section)

==Transformations==
[[File:FourGeometryTransformations.svg|thumb|400px|<div class="center">a) y = f(x) = <nowiki>|</nowiki>x<nowiki>|</nowiki> {{spaces|5}}  b) y = f(x+3) {{spaces|5}} c) y = f(x)-3 {{spaces|5}} d) y = 1/2 f(x)</div>]]

Transformations are applied to a parent function to turn it into a new function with similar characteristics.

The graph of <math>R(x,y)</math> is changed by standard transformations as follows:
* Changing <math>x</math> to <math>x-h</math> moves the graph to the right <math>h</math> units.
* Changing <math>y</math> to <math>y-k</math> moves the graph up <math>k</math> units.
* Changing <math>x</math> to <math>x/b</math> stretches the graph horizontally by a factor of <math>b</math>.  (think of the <math>x</math> as being dilated)
* Changing <math>y</math> to <math>y/a</math> stretches the graph vertically.
* Changing <math>x</math> to <math>x\cos A+ y\sin A</math> and changing <math>y</math> to <math>-x\sin A + y\cos A</math> rotates the graph by an angle <math>A</math>.

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 <math>y=1/x</math> 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 <math>y=f(x)</math>, then it can be transformed into <math>y=af(b(x-k))+h</math>. In the new transformed function, <math>a</math> 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 <math>a</math> values, the function is reflected in the <math>x</math>-axis. The <math>b</math> value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like <math>a</math>, reflects the function in the <math>y</math>-axis when it is negative. The <math>k</math> and <math>h</math> values introduce translations, <math>h</math>, vertical, and <math>k</math> horizontal. Positive <math>h</math> and <math>k</math> 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 <math>R(x,y)</math> is a relation in the <math>xy</math> plane.  For example,
<math display="block">x^2+y^2-1=0</math>
is the relation that describes the unit circle.