Transforming Functions

There are numerous ways to apply transformations to functions to create new functions.
Let's look at some of the possibilities.

Reflections and Functions:

Reflection over the x-axis
 

A reflection is a mirror image.  Placing the edge of a mirror on the x-axis will form a reflection in the x-axis.  This can also be thought of as "folding" over the x-axis.

If the original (parent) function is y = f(x), the reflection over the x-axis gives function h(x), where h(x) = -f(x).

 
Reflection over the y-axis
 
A reflection is a mirror image.  Placing the edge of a mirror on the y-axis will form a reflection in the y-axis. This can also be thought of as "folding" over the y-axis.

If the original (parent) function is y = f(x),
 the reflection over the y-axis gives
 function g(x), where g(x) = f(-x).

 

 

Translations and Functions:  A translation is a sliding of the graph.
Slide to the right or left
 
If the original (parent) function is y = f(x), the translation (sliding) of the function horizontally to the left or right is given by the function g(x),
where g(x) = f(x - b).
  • if b > 0, the graph translates (slides)
    to the right.
  • if b < 0, the graph translates (slides)
    to the left.

Remember that you are subtracting the
value of b from x.
Slide upward or downward
 
If the original (parent) function is y = f(x), the translation (sliding) of the function vertically upward or downward is given by the function g(x), where g(x) = f(x) + b.
  • if b > 0, the graph translates (slides)
    upward.
  • if b < 0, the graph translates (slides)
     downward.

Remember that you are adding the value of b to the y-values of the function.

 

Stretch or Compress Functions:
Horizontal Changes
 
A horizontal stretching is the stretching of the graph away from the y-axis.
A horizontal compression is the squeezing of the graph towards the y-axis.

If the original (parent) function is y = f(x), the horizontal stretching or compressing of the function is given by the function g(x), where g(x) = f(bx).

  • if 0 < b < 1 (a fraction), the graph is stretched horizontally by a factor
    of b units.
  • if b > 1, the graph is compressed horizontally by a factor of b units.
  • if b should be negative, the horizontal compression or horizontal stretching of the graph is followed by a reflection of the graph across the y-axis.
Vertical Changes
 
A vertical stretching is the stretching of the graph away from the x-axis.
A vertical compression is the squeezing of the graph towards the x-axis.

If the original (parent) function is y = f(x), the vertical stretching or compressing of the function is given by the function g(x), where g(x) = bf(x).

  • if 0 < b < 1 (a fraction), the graph is compressed vertically by a factor
    of b units.
  • if b > 1, the graph is stretched vertically by a factor of b units.
  • If b should be negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis.


 


Roberts