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 different vertical transformations or two different horizontal transformations can result in differing functions depending on the order they are applied in.

Let $f(x)=x^{3}+x^{2}$ for the following examples:

• Vertical shift and horizontal reflection: A 3 unit vertical shift, THEN a horizontal reflection of $f(x)$, will result in the function $g(x)=(-x)^{3}+(-x)^{2}+3=-x^{3}+x^{2}+3$. If we apply the horizontal reflection first instead, we will still get the same result.
• Vertical shift and vertical reflection: A 3 unit vertical shift, THEN a vertical reflection of $f(x)$, will result in the function $g(x)=-(x^{3}+x^{2}+3)=-x^{3}-x^{2}-3$. However, if we apply the vertical reflection before the vertical shift, we will get the function $h(x)=-x^{3}-x^{2}+3$, which is not equal to our previous result $g(x)$.
• Horizontal shift and vertical compression: A 3 unit horizontal shift, then a vertical compression of 2, will result in the function $g(x)=2((x-3)^{3}+(x-3)^{2})$. If we apply the vertical compression first, we get the same result.
• Horizontal shift and horizontal compression: A 3 unit horizontal shift, then a horizontal compression of 2, will result in the function $g(x)=(2x-3)^{3}+(2x-3)^{2}$. However, if we apply the horizontal compression first, we get the function $g(x)=(2(x-3))^{3}+(2(x-3))^{2}=(2x-6)^{3}+(2x-6)^{2}$, which is not the same as our previous result $g(x)$.

Resources

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