A transformation takes a basic function and changes it slightly with predetermined methods.  This change will cause the graph of the function to move, shift, or stretch, depending on the type of transformation.  The four main types of transformations are translations, reflections, rotations, and scaling.

Translations

A translation moves every point by a fixed distance in the same direction. The movement is caused by the addition or subtraction of a constant from a function.  As an example, let $f(x) = x^3$.  One possible translation of $f(x)$ would be $x^3 + 2$.  This would then be read as, "the translation of $f(x)$ by two in the positive y direction".  

Graph of a function being translated

The function $f(x)=x^3$ is translated by two in the positive $y$ direction (up).

Reflections

A reflection of a function causes the graph to appear as a mirror image of the original function.  This can be achieved by switching the sign of the input going into the function.  Let the function in question be $f(x) = x^5$.  The mirror image of this function across the $y$-axis would then be $f(-x) = -x^5$.  Therefore, we can say that $f(-x)$ is a reflection of $f(x)$ across the $y$-axis.  

Graph of a function being reflected

The function $f(x)=x^5$ is reflected over the $y$-axis.

Rotations

A rotation is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation.  Although the concept is simple, it has the most advanced mathematical process of the transformations discussed.  There are two formulas that are used:

$x_1 = x_0cos\theta - y_0sin\theta \\ y_1 = x_0sin\theta + y_0cos\theta$ 

Where $x_1$and $y_1$are the new expressions for the rotated function, $x_0$ and $y_0$ are the original expressions from the function being transformed, and $\theta$ is the angle at which the function is to be rotated.  As an example, let $y=x^2$.  If we rotate this function by 90 degrees, the new function reads:

 $[xsin(\frac{\pi}{2}) + ycos(\frac{\pi}{2})] = [xcos(\frac{\pi}{2}) - ysin(\frac{\pi}{2})]^2$

Scaling

Scaling is a transformation that changes the size and/or the shape of the graph of the function.  Note that until now, none of the transformations we discussed could change the size and shape of a function - they only moved the graphical output from one set of points to another set of points.  As an example, let $f(x) = x^3$.  Following from this, $2f(x) = 2x^3$.  The graph has now physically gotten "taller", with every point on the graph of the original function being multiplied by two. 

Graph of a function being scaled

The function $f(x)=x^3$ is scaled by a factor of two.