Function transformations

Let us imagine that initially we have a graph of any function and then we observe that it is translated from position (either horizontally or vertically), compressed (or dilated) and reflected around an axis. The application of these previous processes to any function is called transformation of functions.

transformation of functions

When we talk about function transformations, we apply three fundamental processes:

  1. Translation: that allows us to move a function to the left, right, up or down.
  2. Escalationor: that allows us to widen or narrow a function in a certain direction.
  3. Reflection: that allows us to reflect the function with respect to the x or y axes.

A general expression for transformation of functions is the following:

f (x) = a (xh) + k

Where, a, h and k are real numbers, x the variable.

  • h Represents a horizontal translation of the function.
  • k Represents a vertical translation of the function.

For a there are different cases:

  • If a multiplies the function of x (a · f (x)), then it will expand or compress with respect to the vertical axis:
    • a> 0 will expand.
    • a <0 will be compressed.
  • If a directly multiplies the variable x (a · x), then it will expand or compress with respect to the horizontal axis.
    • a> 0 will be compressed.
    • at <0 it will dilate.

For a reflection on the x-axis to occur, the entire function must be multiplied by some negative number (- f (x)), while for the reflection to occur on the y-axis, it must be multiplied by some negative number on the x-axis. variable x (-x).

Example: We have the following function:

transformation of functions1
Let's see what happens if we add some number to it:

transformation of functions3
The function has been moved vertically. Now let's subtract a number from the variable x:

transformation of functions4
The function has been moved horizontally. Finally, let's multiply by a negative number:

transformation of functions5
The function was reflected with respect to the x axis and expanded with respect to the y axis.