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