Recall our previous work with transformations, where we studied how the graph of the function defined by is related to the graph of , where , , , and are real numbers with . Because such transformations can shift and stretch a function, we are interested in understanding how we can use transformations of the sine and cosine functions to fit formulas to circular functions.

Note in the exercise above several patterns. Given a function , the amplitude of the function was , and the midline of the function was . In addition, each function above had period , unchanged from the period of . We will see later that this is related to the fact that there were no horizontal stretches or compressions of .

Shifts and vertical stretches of the sine and cosine functions

We know that the standard functions and are circular functions that each have midline , amplitude , period , and range . This suggests the following general principles.

In the figure below, we see how the overall transformation comes from executing a sequence of simpler ones. The original parent function (in red) is first shifted units right to generate the blue graph of . In turn, that graph is then scaled vertically by to generate the orange graph of . Finally, the orange graph is shifted units vertically to result in the final graph of in black.

It is often useful to follow one particular point through a sequence of transformations. In the above figure, we see the red point that is located at on the original function , as well as the point that is the corresponding point on under the overall transformation. Note that the point results from the input, , that makes the argument of the cosine function zero: .

While the sine and cosine functions extend infinitely in either direction, it’s natural to think of the point as the “starting point” of the cosine function, and similarly the point as the starting point of the sine function. We will refer to the corresponding points and on and as anchor points. Anchor points, along with other information about a circular function’s amplitude, midline, and period help us to determine a formula for a function that fits a given situation.

The previous example illustrates a general technique for giving the formula of the graph of a circular function of period with a given midline, amplitude, and -intercept. Say is a circular function with period . If the midline is , the amplitude is , and we know that , then we have a procedure to follow to produce a formula for .

(a)
Stretch the graph vertically by a factor of to obtain
(b)
Shift the graph vertically up by units to obtain
(c)
Find such that
(d)
Shift the function horizontally so that the point transforms into .

Note that this procedure can work with as well as . Let’s see an example.

Horizontal scaling

There is one more very important transformation of a function that we’ve not yet explored in the trigonometric context. Given a function , we want to understand the related function , where is a positive real number. The sine and cosine functions are ideal functions with which to explore these effects; moreover, this transformation is crucial for being able to use the sine and cosine functions to model phenomena that oscillate at different frequencies.

By using a graphing utility such as Desmos, we can explore the effect of the transformation , where .

By experimenting with the slider, we gain an intuitive sense for how the value of affects the graph of in comparision to the graph of . When , we see that the graph of is oscillating twice as fast as the graph of since completes two full cycles over an interval in which completes one full cycle. In contrast, when , the graph of oscillates half as fast as the graph of , as completes only half of one cycle over an interval where completes a full one.

We can also understand this from the perspective of function composition. To evaluate , at a given value of , we first multiply the input by a factor of , and then evaluate the function at the result. An important observation is that This tells us that the point lies on the graph of since an input of in results in the value . At the same time, the point lies on the graph of . Thus we see that the correlation between points on the graphs of and (where ) is We can therefore think of the transformation as achieving the output values of twice as fast as the original function does. Analogously, the transformation will achieve the output values of only half as quickly as the original function.

Recall that given a function and a real number , the transformed function is a horizontal stretch of the graph of . Every point on the graph of gets stretched horizontally to the corresponding point on the graph of . If , the graph of is a stretch of away from the -axis by a factor of ; if , the graph of is a compression of toward the -axis by a factor of . The only point on the graph of that is unchanged by the transformation is .

Circular functions with different periods

Because the circumference of the unit circle is , the sine and cosine functions each have period . Of course, as we think about using transformations of the sine and cosine functions to model different phenomena, it is apparent that we will need to generate functions with different periods than . For instance, if a ferris wheel makes one revolution every minutes, we’d want the period of the function that models the height of one car as a function of time to be . Horizontal scaling of functions enables us to generate circular functions with any period we desire.

We begin by considering two basic examples. First, let and . We know from our most recent work that this transformation results in a horizontal compression of the graph of by a factor of toward the -axis. If we plot the two functions on the same axes, it becomes apparent how this transformation affects the period of .

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Figure 1: A plot of the parent function, (dashed, in gray), and the transformed function (in blue).

From the graph, we see that oscillates twice as frequently as , and that completes a full cycle on the interval , which is half the length of the period of . Thus, the “” in causes the period of to be as long; specifially, the period of is .

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Figure 2: A plot of the parent function, (dashed, in gray), and the transformed function (in blue).

On the other hand, if we let , the transformed graph is stretched away from the -axis by a factor of . This has the effect of doubling the period of , so that the period of is , as seen in the previous figure.

Our observations generalize for any positive constant . In the case where , we saw that the period of is , whereas in the case where , the period of is . Identical reasoning holds if we are instead working with the cosine function. In general, we can say the following.

For any constant , the period of the functions and is

Thus, if we know the -value from the given function, we can deduce the period. If instead we know the desired period, we can determine by the rule .

You might wonder why we’ve chosen to use the formula with the parentheses instead of without the parentheses. In the former formula, we save the horizontal shift until the end, but in the latter, we do the horizontal shift before anything else. The reason we prefer the former is that the midline, amplitude, and period of a function are easy to plug into the formula, and plugging these in results in a formula of the form . We can then find the horizontal shift we need to apply to the graph of to obtain .

You may have noticed that the last example involved a reflection, a transformation we have not talked about in the context of circular functions. This is because reflections of circular functions turn out to be obtainable by shifts, so when we’re given graphical information and asked to reconstruct the formula for a function, we don’t need to use reflections. Reflections may show up when we are given a formula and asked to provide a graph. But we already have lots of practice producing graphs in this situation and can draw on this prior experience with transformations.