The goal of this lesson is to apply transformations to different functions in order to model a set of data. Students examine the key function types that they have studied up to this point: polynomials, rationals, radicals, and exponentials. In each case, once the basic shape of the function type is chosen, vertical and horizontal translations, scalings, and reflections are applied in order to model the data as well as possible.
It turns out that each function type can be made to fit the data quite well. In fact, a quadratic function seems to give the best fit. Because the data is for the temperature of a bottle of water which has been removed from the refrigerator, the context also plays an important role in choosing the best model. The water will continue to warm, eventually stabilizing at room temperature. A quadratic model, however, will predict the warming but then it eventually reaches its maximum and decreases afterward. From this perspective, the exponential model and rational function model match the long term data trend better.
In this lesson, students model temperature data with different functions (MP4). While the data is given, only a partial model (the general function type) is included and students need to build an accurate model from here by applying transformations. This can be done by hand via experimenting, but students may also choose to use graphing technology to help choose the appropriate translations, scalings, and reflections (MP5). For students who do not use technology to find the transformations, they will need to think about the shape of the function and the structure of each translation or scaling in order to find an appropriate combination (MP7). Deciding which model is "best" requires critical analysis not only about how well each model fits the data but also about the behavior of each model as time continues to elapse (MP3).
- Compare and contrast different models for the same data.
- Use transformations to create a function that models data given a simple starting function.
Let’s model with functions.
- I can transform a function so its graph models a data set.