Composition and Transformations
One way to visualize composition is to decompose into and and plot the graph of each in its respective coordinate planes. You can then follow the arrows: Plug into to get ; then plug into to get ; and finally match up and to get a point on . The inside function depends on a parameter , which may be changed. Equivalently, this construction is the projection onto the - plane of the intersection of the surfaces and .
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