Lesson 17

The purpose of this activity is for students to consider the graphs of three familiar types of functions, radical, linear, and quadratic, and understand that when the same transformations are applied to different function types, the graphs change in the same way. This warm-up also directly addresses that the order in which transformations are applied can matter, something which comes up repeatedly throughout the lesson.

Arrange students in groups of 2.  Give students brief quiet work time followed by sharing their thinking with a partner.

If students have difficulty identifying the transformations with precision, encourage them to try marking specific points on the graphs and considering how transformations take the point from one graph to the corresponding point on the other graph.

There are many possible transformations that take the upper graph to the lower graph. Begin the discussion by displaying the three graphs and selecting students to share their transformation steps.

If the same set of steps is suggested for all three transformations, draw students' attention to them. Otherwise ask “What happens if we reflect the upper graph across the \(x\)-axis, and then translate it left 1 and down 3?” (That transformation works for all three.) Next, ask students if the transformation would work if we did the steps in a different order. (No, translating down and then reflecting across the \(x\)-axis has a different result than reflecting across the \(x\)-axis and then translating down.)

This info gap 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 which 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 info 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 ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Tell students they will continue to work on identifying transformations between two functions. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Provide students with access to small sheets of graph paper.

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 \(Q(x) = 4\sin(2x) + 14\). For problem card 2, if the horizontal stretch is done first and then the horizontal translation the result would be \(S(x) = \text-\cos(\pi (x + \pi))+3\)
• For the first card, the vertical translation will take \(\sin(2x) + 3\) to \(\sin(2x )+ 5\). For the vertical stretch, make sure students understand that both \(\sin(2x)\) and \(5\) 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 \(\sin(2x) + 5\).
• For the second card, the horizontal translation changes \(\text-\cos(x) + 3\) to \(\text-\cos(x+\pi) + 3\) and then the horizontal scaling by \(\frac{1}{\pi}\) gives \(\text-\cos(\pi x + \pi) + 3\). 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.

This activity is optional. 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.

Arrange students in groups of 2. Make sure each group has access to graphing technology.

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 \(f\) and \(g\) for all to see. Invite groups to share their transformations, recording them alongside the graphs listing as many as students were able to identify. Here are some questions for discussion:

• “Why do \(g(x) = 2f\left(2(x-\frac{\pi}{4})\right)+3\) and \(g(x) = 2f\left(2x-\frac{\pi}{2}\right)+3\) 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.)
• “\(g(x)=\text-2f(2x+\frac{\pi}{2})+3\) works. How would you describe this transformation in words?” (Translate the graph of \(f\) left \(\frac{\pi}{2}\), stretch horizontally by a factor of \(\frac{1}{2}\), stretch vertically by a factor of 2, reflect over the \(x\)-axis, then translate up 3.)