Lesson 11

This warm-up presents decontextualized data that students will analyze in the lesson. The data is plotted in the coordinate plane and the \(y\)-values grow at a slower rate as the \(x\)-values increase. Students brainstorm functions that they have seen which match this overall behavior. In the main activity of the lesson, students apply transformations to each type of function which they identify here, trying to model the data as well as possible.

By first considering the data without context, students make sense of the shape of the data in a constructive way that will be integral to work building a model (MP1).

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graph. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the rate of growth of the points did not come up, ask students to discuss this idea and articulate that the \(y\)-values increase as \(x\)-values increase, but the rate of increase slows down as \(x\)-values grow.

Next, ask students, “What functions have you seen that have this general shape?” Possible function types students have studied in depth to this point include:

• polynomials (for example, a parabola opening downward, \(y = \text-x^2\))
• rational functions (for example, \(y = \text-\frac{1}{x}\))
• radical functions (for example, \(y = \sqrt{x}\))
• exponential functions (for example, \(y = \text-2^{\text-x}\))

In the warm-up, students brainstormed different types of functions whose graphs have a particular shape. In this activity, they work with those function types to try to model the temperature of a bottle of water after it has been removed from the refrigerator. They will apply different transformations to 2 possible functions, including translations, scale factors, and reflections, to model the data as well as possible, engaging fully in the making a model aspect of mathematical modeling (MP4).

Next, students look at the set of models and consider which one is best to model with situation. In addition to fitting the data well, they also consider end behavior of the functions and of the temperature of the water bottle.

Monitor for groups using different strategies to determine the parameters for their functions. For example, some groups may use technology to quickly try many options for a vertical scale factor while others may first make a quick calculation using a specific point.

If time allows, consider having groups present their strategies for creating a model to fit the data.

There are many paths to fitting a function to data. If groups are not sure what type of transformation to start with, suggest they focus on transformations that will get the function to match the same general shape as the data without worrying about fitting the data exactly.

For groups who work with the exponential model \(f\) given by \(f(x) = 2^x\), they will need to replace \(x\) with \(\text-x\) and apply a reflection across the vertical axis as well. If they struggle, consider suggesting that they start with \(f(x) = \left(\frac{1}{2}\right)^x\), which has the right general shape but needs to be reflected across the vertical axis and translated.

Begin the discussion by selecting previously identified groups to share how they determined their function via transformations. For example, for the quadratic model \(Q(x) = 70 - 1.4(x-5)^2\), ask students questions like:

• “How does the 5 in the expression \((x-5)^2\) influence the graph?” (The 5 translates the vertex of the parabola right to \(x = 5\).)
• “How does the coefficient 1.4 influence the graph?” (1.4 is a vertical scale factor which determines how "steep" the parabola is.)
• “How does the negative in front of the 1.4 influence the graph?" (The negative reflects the parabola over the \(x\)-axis so that it opens downward.)
• “How does the constant 70 influence the graph?” (By using a vertical translation of 70, the \(y\)-value of the vertex is moved from \((5,0)\) to \((5,70)\).)

Where possible, ask students to reason about specific aspects of their models in terms of the context. For example:

• In the exponential, quadratic, and rational sample response models, the number 75 or 70 is the temperature that the water approaches as time goes on, that is the room temperature.
• In the quadratic model, the number 5 indicates that the vertex of the graph is at \(x = 5\). In the context, this means that according to the quadratic model, the temperature of the water will begin to decrease after 5 hours.
• The given data starts at 0 hours and ends after about 5 hours, so as far as matching the data goes, a domain of \(0 \leq x \leq 5\) is reasonable. Larger values of \(x\) could be considered to see what the model predicts as time goes by, but negative values of \(x\) are not interesting for the model since that is before the water was removed from the refrigerator.