In this activity, students sort different graphs and equations. A sorting task gives students opportunities to analyze representations and structures closely and make connections (MP7).

Monitor for different ways groups choose to categorize the graphs and equations, but especially for categories that distinguish between the transformations that were applied to each function. 

The purpose of this discussion is for students to identify transformations from graphs and equations of functions. Select groups to share their categories and how they sorted their graphs and equations. Discuss as many different types of categories as time allows, but ensure that one set of categories distinguishes between the transformations applied to each function. Attend to the language that students use to describe their categories, graphs, and equations, giving them opportunities to describe the transformation more precisely. Highlight the use of terms like "horizontal stretch by a factor of ," "reflection," "input," and "output." 

If no students identify the two transformations that were applied to the functions, invite them to discuss this idea now. (One set of graphs and equations has undergone this sequence of transformations: Reflect across -axis, translate up 3, horizontal stretch by a factor of 5. The other set of graphs and equations has undergone this sequence of transformations: vertical stretch by a factor of 3, translate left 5.) Invite students to share how they identified these transformations from the graph or from the equation.

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

The purpose of this activity is for students to leverage repeated reasoning to generalize about the effect of a transformation on the equation of a function (MP8). They apply the same transformation to different function types and analyze the resulting equations. Students then apply the patterns they have noticed in the equations to a general equation for the transformation of an unknown function. 

If students are unsure how to write an equation for when the function is unknown, consider saying:

• “Tell me more about how you wrote your equations for the four known functions.”

• “How can thinking about the transformations made to the input and the output of the function help you write an equation for in terms of ?”

The purpose of this discussion is for students to generalize how transformations of functions affect their equations. Here are some questions for discussion:

• "How were the resulting equations similar?" (The horizontal stretch meant there was a coefficient of with the in each equation. Shifting left by 1 meant that each function had in the input, so it was in the parenthesis with the in the equation. The vertical reflection and stretch affected the output of the function, so all of the equations were multiplied by -4 outside of the parentheses.)
• "How were the resulting equations different?" (For the and equations, the only difference was the exponent. For the and equations, the -4 coefficient had to go outside the operation. The function looked really different because the input was in the exponent of the equation, and the output was not.)
• "How can we see the transformations of a function from a general equation ?" (Changes to the input will be inside the parentheses with the , and changes to the output will be outside of the .)