1.2. Graphical Transformations http://www.ck12.org
When transforming a function, you can transform the argument (the part inside the parentheses with thex), or the
function itself. There are two ways to linearly transform the argument. You can multiply thexby a constant and/or
add a constant to thexas shown below:
f(x)→f(bx+c)The function itself can also be linearly transformed in the same ways:
f(x)→a f(x)+dEach of the lettersa,b,c,anddcorresponds to a very specific change. Some of these changes are straightforward,
while others may be the opposite of what you might expect.
- ais a vertical stretch. Ifais negative, there is also a reflection across thexaxis.
- dis a vertical shift. Ifdis positive, then the shift is up. Ifdis negative, then the shift is down.
When transforming the argument of the function things are much trickier.
-^1 bis a horizontal stretch. Ifbis negative, there is also a reflection across theyaxis.
- cis a horizontal shift. Ifcis positive, then the shift is to the left. Ifcis negative, then the shift is to the right.
Note that this is the opposite of what most people think at first.
The trickiest part with transforming the argument of a function is the order in which you carry out the transforma-
tions. See Example A.
Example A
Describe the following transformation in words and show an example with a picture:f(x)→f( 3 x− 6 )
Solution:Often it makes sense to apply the transformation to a specific function that is known and then describe the
transformation that you see.
Clearly the graph is narrower and to the right, but in order to be specific you must look closer. First, note that
the transformation is entirely within the argument of the function. This affects only the horizontal values. This
means while the graph seems like it was stretched vertically, you must keep your perspective focused on a horizontal
compression.