Malin's Magnificent Math

Scaling & Shifting of Graphs

Elementary Waves

Affine Functions

Waves in 2D & 3D

Affine Functions

Affine Functions in 1D:

An affine function is a function composed of a linear function + a constant and its graph is a straight line. The general equation for an affine function in 1D is: y = Ax + c.

An affine function demonstrates an affine transformation which is equivalent to a linear transformation followed by a translation. In an affine transformation there are certain attributes of the graph that are preserved. These include:
  • If three points all belong to the same line then under an affine transformation those three points will still belong to the same line and the middle point will still be in the middle.
  • Parrallel lines remain parrallel.
  • Concurrent lines remain concurrent.
  • The ratio of length of line segments of a given line remains constant.
  • The ratio of areas of two triangles remains constant.
  • Ellipses remain ellipses and the same is true for parabolas and hyperbolas.

Affine Functions in 2D:

In 2D the equation of an affine function is f(x,y)=Ax + By + C.

The graph of a wave in 2D as shown in the next section shows the example of a graph of a 2D affine function.

Affine Functions in 3D:

In 3D the equation of an affine function is f(x,y,z)=Ax + By + Cz + D.