If students have difficulty identifying the transformations with precision, consider saying:

• “Describe the transformations you see in your own words.”

• “Mark corresponding points on the two graphs. How could those two points help you determine the transformations?”

This activity gives students an opportunity to determine and request the information needed to find and graph the transformation of a given function. In each case, the given function is a trigonometric function that has been translated horizontally or vertically and also reflected or scaled. The problem card gives the graph and equation of the trigonometric function. The student with the problem card figures out which transformations to apply to the graph and equation. The data cards give multiple ways for students to figure out the transformation. Monitor for students who:

• Ask for the explicit transformations and the order in which they are applied.
• Ask for the amplitude, midline, and period of the new function.

The Information Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Focus on these points:

• The order of the transformations matters. For Problem Card 1, if the vertical stretch is done before the vertical translation it gives a different result, namely . For Problem Card 2, if the horizontal stretch is done first and then the horizontal translation is done, the result would be .
• For the first card, the vertical translation will take to . For the vertical stretch, make sure students understand that both and need to be scaled by 4. The vertical stretch by 4 multiplies the output of the function by 4 and in this case the output, after the translation, is .
• For the second card, the horizontal translation changes   to   and then the horizontal scaling by gives  . Unlike Problem Card 1, the scaling does not influence the constant 3 because the scale is horizontal, while the 3 translated the graph vertically. 

Ask students if they considered asking for the amplitude, period, and midline. In a sense, this information may be more straightforward to process because the order of the transformations is already encoded in this information. But the student with the problem card may not realize that this information is available. 

Finally, ask students if they considered sketching the graph first and using this to find the transformed function. The sketch can help show major features of the graph (amplitude, midline, period) and serve as a guide for finding a valid equation.

Use this activity to give students extra practice using technology to transform periodic functions.

The goal of this activity is for students to use graphing technology to transform the graph of a function to match another function. Because trigonometric functions are periodic, there are many ways to make the given transformations. Having the graphs allows students to use their visual intuition to visualize the transformations, and then the main work of the activity is translating this into function notation.

The goal of this discussion is for students to make connections between the different ways to think about a graphical transformation and the different ways to algebraically rewrite an expression for a function.

Display the graphs of and . Invite groups to share their transformations, recording them alongside the graphs and listing as many as students were able to identify. As students share their transformations with the class, call students’ attention to the different ways the vertical and horizontal translations and vertical and horizontal stretches are represented in the equations. If possible, use technology to demonstrate the transformations as students share. Here are some questions for discussion:

• “Why do  and both result in the same graph?” (The second is what happens if you use the distributive property. The value of the expression doesn't change, just the look.)
• “ works. How would you describe this transformation in words?” (Translate the graph of  left , stretch horizontally by a factor of , stretch vertically by a factor of 2, reflect over the -axis, then translate up 3.)