Problem: Using the knowledge of transformations guess the graph of the following functions:
Solution: As you can guess, this problem is "educational", we will try to make some point here. Since the only difference between the first two problems is the placement of the absolute value, it seems that we will focus on the order of transforms and the effect of mirroring here. Let's start with the first example.
a) Clearly the basic function here is the exponential.
We are applying two transformations to the argument, in the order of
evaluation we first subtract 1 and then take the absolute value. Since
transformations to argument should be applied "last to first", we should
start with the graph of
x → |x| → |x − 1|.
This fits, so we do it.
One quick check that we did not make any major screw-up: The function as
given has a suspicious breaking point when the absolute value is zero, that
is, when
b) Now the second function. There the order is the opposite, when we evaluate the function, we first do the absolute value and then subtract, so the transformations should go "shift-mirroring". Quick check using the replacement:
x → x − 1 → |x| − 1.
This fits, let's go.
Quick check: The break in the absolute value happens when
c) The third function is also based on the exponential.
In fact we can start from the graph we just drew, then apply two more
transforms to it. The function
Obviously, here it helps if we have some better idea concerning just what
parts of the shifted exponential will be below the x-axis. In other
words, we want to know which parts of the above graph are below the level