Trigonometry is integral to mathematics education. The field of trigonometry plays a crucial role in the study of mathematics and its applications. Despite the importance of the subject, students struggle to understand trigonometric constructs such as angle measure. It has also been noted how students struggle to understand transformations of functions generally. Our review of the literature found few studies specifically on students’ understanding of transformations of trigonometric functions, but evidence exists showing students have difficulties with the concept. Here, a MATLAB program called

Trigonometry is integral to both pure and applied mathematics education (^{ix}

Here, a tool is presented for teaching transformations of trigonometric functions. The goal of this tool is to help students understand properties of function transformations rather than memorize them.

A transformed sinusoid represented in TrigReps.

While additive and multiplicative horizontal transformations are individually counterintuitive, their combination is also counterintuitive. The order of operations states that multiplication is performed before addition, but additive horizontal transformations must be performed before multiplicative ones.

In addition to the literature review, a case study is presented examining how

In order to determine how

How do students interact with

How (if at all) does

In this section, a review of the literature is presented indicating that students struggle with both trigonometry and graphical transformations. Previous studies have shown that students who are unable to change between multiple representations have difficulty in trigonometry (

The existing literature on learning trigonometry raises several concerns. Numerous studies have found that students leave trigonometry classrooms with poor understanding of the subject. Studies have depicted struggles among secondary students (

Given a function

Apart from trigonometry, numerous researchers have stated that students tend to experience difficulty learning graphical transformations generally (

Strategies have been suggested to help students justify the nuances of graphical transformations instead of memorize them (

The coordinate axes and the curve as separate objects (

Hall and Giacin (

The horseshoe method shifting the function ^{2} three units to the left (

Students learning horizontal transformations may be aided by utilizing MERs. Weber (

In contrast, dynamic and interactive representations have been effective in trigonometry classrooms (

One limitation of using technology in the classroom is that students may attempt to use it to replace actively thinking about the mathematics (

A pilot study for

To examine the efficacy of

The final two shifts of attention are more active and time-consuming. Once students have enough details, they can begin noticing patterns and testing hypothesized properties. This is what Mason (

Once students have been introduced to the mathematical tasks, they have begun holding wholes. Using

Bornstein’s (

This qualitative case study examines the work of a group of three students in an undergraduate precalculus class at a large, northeastern university. The group consisted of three women, Alexa, Brianna, and Caitlin (the names given here are pseudonyms). Each of the three were freshmen. Alexa and Brianna were majoring in business administration, while Caitlin was studying marine biology. The class was presented with a fifty-minute lecture covering transformations of trigonometric functions. The following class, students were assigned a worksheet to complete in groups of three or four. Each group was provided with a laptop computer already powered on and running

The students’ group work was transcribed and coded. The initial codes noted which representations students were given, which representations they used, whether they used ratio or unit circle definitions of trigonometric functions, if they were incorrect, and the particular strategies they employed. Common strategies included using reference angles on the unit circle, memorized outputs of trigonometric functions, the Pythagorean identity ^{2}(θ) + ^{2}(θ) = 1, and the special 30-60-90 and 45-45-90 right triangles.

The data was additionally coded according to Mason’s (

Students often used details that they did not discern from the representations given to them. When students utilized their prior knowledge, it was coded as

The code

Alexa, Brianna, and Caitlin were able to successfully complete the set of tasks that examined single transformations. However, there was not enough time for them to begin the set of tasks that examined combinations of transformations. Discussion amongst the group showed that the students made predictions regarding function transformations, used

Part of the first task involved finding a function with twice the amplitude of

For twice the amplitude, do we just do 2

That made the amplitude greater. The peaks are taller.

Why isn’t it dinging though?

Because the frequency is still the same. It needs to have a lot more of the loop-dee-loops in the same 2π. I’m just going to write

It’s the amplitude.

Later, when they were examining the effects of transformations on the audio representation, they began with input

That’s a cool graph.

Why does it look like that?

It made a sound at the beginning like when you plug a guitar into an amp…. What if we did amplitude as well as [input

Aaaaaaah!

That did something. Let’s not try 100. Let’s go back to 10. We did an amplitude of 100 and a frequency of 400. We won’t do that anymore…. I guess that’s why it’s called an amp.

The group noticed that when they changed the

The group spent very little time on the set of tasks examining vertical shifts. While looking for a function that shifted

Horizontal shifts were slightly more difficult. While trying to shift the graph of

Did that make it go to the right? So plus?

It would be

…and to the right by seven. Why did that shift?

What?

The radius went to a different spot.

Although Brianna commented on the shift of the radius, the group did not explore that phenomenon, and it was not remarked upon in their written work.

The next task asked them to find a function with triple the frequency of

This group was able to complete the tasks that addressed each of the individual transformations: vertical stretches, vertical shifts, horizontal stretches, and horizontal shifts. The group did not have enough time to finish the activity. The remainder of the tasks examined combinations of transformations. An additional class period would have been necessary for students to work through these concepts.

Recall Mason’s (

Details discerned by the group include descriptions of representations such as when Caitlin said “That made the amplitude greater. The peaks are taller.” Also included were observations like “I heard that a lot clearer,” and “The radius went to a different spot.”

The group needed to recognize some relationships in order for

There is evidence that the group perceived a number of properties for single transformations. Given

While their work is consistent with having learned that multiplicative horizontal transformations behave counterintuitively, they referred to these transformations in terms of the frequency of the function, rather than the period. Since the frequency is the inverse of the period, the terminology is not counterintuitive; instead of saying that multiplying by large numbers shrinks the function, they said that it increases the frequency. Because of this, they never commented on the difference between vertical multiplicative transformations and horizontal ones. Finally, the group did not have time to examine combinations of transformations.

The final stage of Mason’s (

Each task had multiple parts so that students could make and test hypotheses. After finding that 2

It cannot be determined exactly which of the learning goals were newly perceived properties and which were recognized relationships that the students were familiar with to start the activity. That being given, by the end of the activity, the students demonstrated that they could predict the behavior of single transformations.

Out of the seven learning goals for function transformation, the group demonstrated that they had learned at least five. They showed that they understood the effects of multiplicative transformations and additive transformations. They also showed that they understood the effects of transformations inside and outside of the parentheses. Additionally, the group noted that additive horizontal transformations behave counterintuitively. There is not enough evidence to show that they understood the counterintuitive behavior of multiplicative horizontal transformations, and the group did not examine combinations of transformations.

This study demonstrated that

The audio representation provides students with another way to connect their experiences with the mathematics. Students are familiar with the concepts of sound volume and frequency, and

In addition to providing another representation for students to connect with, the audio representation could affect students’ motivation. In this study, group members expressed excitement about the audio representation. Previous researchers have indicated that they believe students are more motivated learning trigonometry with auditory applications than with pure mathematics (Kessler 2013;

This study was limited by its short implementation period. Future studies of trigonometric transformations using

The author has no competing interests to declare.