Translations, reflections and symmetry.

A translation maps each point in an object with another new point by an addition or subtraction rule. This has the effect of "sliding" an object from one location to another without changing its size. In the example below, figure A is translated to a new position via the rule: x’ = x +3 and y’ = y +20 The new figure is the same shape and size as the original.

Translation of A to A’

Functions can be translated in a similar fashion. To add translate a given function we want each (x, y) mapped to (x+3, y+20) . For a given function y = f(x), y’ = f(x-3)+20 will achieve this.

Another transformation that retains the original shape of the object is a reflection. A reflection flips all the points in the object over a line. This line is called the line of reflection. The original and reflected figures are mirror images. We say these objects are symmetric with respect to a line, L if they are the reflections of each other across L. Some examples of reflections and symmetric figures follow,

Reflection across the y-axis

A function f(x) is reflected across the y-axis by letting x’ = -x or y’ = f(-x).

Example: y = 3x + 1 and y’ = 3(-x) + 1
 
 

y = 3x + 1 and y = 3(-x) + 1 are symmetry across the y axis.

Reflection across the x-axis

A function f(x) is reflected across the x-axis by letting y’ = -y or y’ = -f(x)

Example: y = 3x + 1 and y’ = -(3x+1)

y = 3x + 1 and y’ = -(3x+1) are symmetric across the x- axis.

Reflection across the line y=x

A function is reflected across the line y=x by letting y’= x and x’= y or

y’ =f-1(x). The inverse function, f-1(x) can be found by solving the equation y = f(x) for x and then let y’ be the resulting expression.

Example: Reflect y = 2x + 1 across the line y = x.


 
 

y = 2x + 1 and y = (y-1)/2 are symmetric across the line y = x.

More on symmetry:

A test for symmetry about the x-axis is: Does f(x)=-f(x)?

A test for symmetry about the y-axis is: Does f(x)=f(-x)?