Choose functions for
and
. The first two choices for
scale and translate the graph of
. The third incorporates a reflection. The rest transform the graph of
in complicated ways. Alternately, the outside function
may be viewed as transforming the graph of
. The choices for
and
were selected because of their fundamental nature. In some cases they are scaled or translated for the sake of presentation. While the focus of this Demonstration is composition, studying the composition of these functions is a good way to improve your understanding of these functions.
Sliding
shows how the values of
are determined from the values of
and
. Sliding
show how the transformation of the graph of
changes. This can be helpful for understanding the simple transformations studied in precalculus (the first three choices for
). Highlight "surfaces" to show the intersection of
and
.
Snapshot 1: a scaling—the period of the tangent is stretched
Snapshot 2: a translation—the vertex of the parabola is shifted from
toward the origin
Snapshot 3: the effect of substituting an absolute value into a function—a mirror image about the vertex of the graph of the absolute value
Snapshot 4: the effect of substituting into the absolute value function—the part of the graph below the
axis is reflected over the
axis
Snapshot 5: the important example from calculus,
—looks better with larger values of
Snapshot 6: the square root of a function whose graph is tangent to the
axis (contact order exactly 1) has a "V" like the absolute value
![[Snapshot]](HTMLImages/index.en/thumbnail_1.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_2.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_3.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_4.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_5.gif)
![[Snapshot]](HTMLImages/index.en/thumbnail_6.gif)
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