Modeling with Trigonometric Functions
(Lesson 6.7)
Unit 6 Day 10
CED Topic(s): 3.7
Learning Targets

Interpret a sinusoidal function's period, amplitude, midline, and range in context.

Construct a trigonometric model based on data points and key features.
Quick Lesson Plan
Experience First:
This final lesson of the unit mirrors the very first lesson of the unit: interpreting (and now constructing) models of periodic phenomena. Students will use everything they have learned in this unit to generate a sinusoidal function that models the temperature outside of Karissa’s house. Before constructing the model, we ask them to identify and interpret key features of the scenario, which is a critical aspect of determining the parameters of the sinusoidal function. In question 1 they determine the amplitude and range of the function. In question 2 they sketch the function. Note that students may not get the concavity quite right when they are sketching at first. Some may sketch their entire graph concave down (resembling the “up bump” of a sinusoidal graph) rather than graphing the entire period of a cosine curve. We hope that by the time students construct the equation, they will automatically update their graph. In question 3 they interpret the average temperature of the day and connect this to the midline of the graph.
In question 4 they are asked to make sense of or interpret the model. Though students’ answers may vary, some might make the argument that it is unlikely that the highest temperature would happen at noon, as the hottest temperature of the day tends to be about 3 PM (depending on where you live!).
Question 8 has students think about what would happen in the next 24 hour period. Since the period is 24 hours, students should see that the exact same temperatures would be reached as on day 1; temperatures would start at the minimum of 70˚, rise to 90˚ and then fall down to 70˚ again. This is a nice review of what students learned at the beginning of the unit about periodic behavior.
Question 9 is a challenge! Since sine and cosine functions are just horizontal shifts of each other, we wanted to see if students could make the adjustments so that their graph was represented by a sine curve. Because the graph has both a horizontal stretch and a horizontal shift, determining the value of c can be tricky. However, substituting in a few values of t can help them validate their model.
Monitoring Questions:

Why is the range of this function relevant? How is this information used? (Think weather reports, predicting high and low temps)

Without any additional information or context, how can you tell if a graph represents the full cycle of the cosine curve or half the cycle of a sine curve?

How will you determine the value of b?

How did you know that the value of a would be negative?

Is it always possible to write a function model using either cosine or sine?
Formalize Later:
Students have already had practice identifying and interpreting the key features of a sinusoidal graph. The main goal of today’s lesson is to be able to use those key features to construct a function model. This requires finding the value of the parameters in y=asin(b(xc))+d or y=acos(b(xc))+d. Finding “a” and “d” is pretty simple for students, while “b” and “c” can be a bit trickier. We will often look at cases where c=0. Remind students that the same strategies they’ve used all year (solving a system of equations to determine parameter values, identifying transformations, etc.) can be used here as well! Often students will be able to determine the values of “a”, “b”, and “d” from the context or by calculating the period and then use an inputoutput pair to solve for c.
In the debrief take a few moments to talk about why a cosine function was chosen as the model (at least initially). Students should be able to articulate that functions that start at their max or min value are easily represented by a cosine function (or its reflection) and not require any horizontal shifts. Functions that start at their average value are easily represented by a sine function (or its reflection) without requiring any horizontal shifts. However, an equivalent model can always be found using the other function.
Question 8 gives you the opportunity to talk about restricted domains, tying back to what students learned in Lesson 3.4 about assumptions and limitations of a model. The sine and cosine functions are defined for all real numbers, but the constraints of a scenario may limit the values for which a particular function is valid. For example, the temperature function students construct in this lesson may be valid for the next day (24≤t≤48), but likely not for many days later.
When looking at modeling contexts for sinusoidal functions, make sure students have the opportunity to go backwards and forwards. They should be able to construct a model given key features of the scenario AND they should be able to determine key features of a scenario from a function model. The Check Your Understanding question gives students a chance to do the latter.