Multiple Transformations of Functions

See Single Transformations of Functions for more information on translating, reflecting, compressing, and stretching functions.

Combining Functions

Vertical and horizontal shift: $f(x)=x^{2}+x(red)$ and $g(x)=(x-3)^{2}+(x-3)+5$ (blue; f(x) shifted 3 units right and 5 units up)

If $f(x)$ is some function, then $g(x)=f(x-h)+k$ is the function $f(x)$ shifted h units horizontally (to the right for h > 0 and to the left for h < 0) and k units vertically (up for k > 0 and down for k < 0). For example, $g(x)=(x-3)^{2}+(x-3)+5$ is the function $f(x)=x^{2}+x$ shifted 3 units to the right and 5 units up. $g(x)={\sqrt {x+2}}-6$ is the function $f(x)={\sqrt {x}}$ shifted 2 units to the left and 6 units down.

The order that two transformations are applied can change the resulting function. A vertical transformation and a horizontal transformation can be applied in either order and will result in the same function. However, two vertical transformations or two horizontal transformations can result in differing functions depending on the order they are applied in.

Let $f(x)=x^{3}$ for the following examples:

• Vertical shift and horizontal reflection: A 3 unit horizontal reflection THEN a 3 unit vertical shift of $f(x)$ will result in the function $g(x)=(-x)^{3}+3$. If we apply the vertical shift first instead, we will still get the result $g(x)=(-x)^{3}+3$.
• Vertical shift and vertical reflection: Let $f(x)=x^{3}$. A 3 unit horizontal reflection THEN a 3 unit vertical shift of $f(x)$

Resources

• Combining Transformations, University of Houston
• Sequences of Transformations, Lumen Learning