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Sketching graphs of f, f’, and f’’ (Topics 5.8-5.9)

Unit 5 - Day 8

​Learning Objectives​
  • Connect features of functions and their derivatives given in graphical, numerical, and analytic representations

​Success Criteria
  • I can determine key features of the graph of f(x) given information about f’(x) and f’’(x) in various representations. 

  • I can sketch graphs of f, f’, and f’’.

Quick Lesson Plan
Activity: 20 Questions (Calculus Style)


Additional Materials:


Lesson Handout


In this activity, students will work in groups of four to practice sketching graphs of functions and their derivatives. One person will be given a graph card and this student is the only one who may look at the graph card during that round. The other three students get to ask 20 questions to ascertain as much information as possible about the graph that they must sketch on their whiteboard. Students practice using precise vocabulary and making connections between graphs of f, f’, and f’’. 

Teaching Tips

Begin the activity by explaining the goal of the activity, the group member roles, and the three rounds of the activity (See 20 Questions Instructions below). Emphasize that the graph holder can only answer with ‘Yes’, ‘No’, or a numerical value. Consider giving examples of questions students may ask about the graph. In Rounds 2 and 3, remind students that they can as the graph holder about the graph on their card OR about the graph they have to sketch (thus forcing the graph holder to do the analysis!)

Walk around and make sure all group members are participating and listen for correct vocabulary. You may need to give hints (either to the graph holder or to the other group members) to help a group along. In Round 3, you may want to prompt students to ask questions about the general shape of the curve (linear, parabolic, etc.).

Possible Modifications:
1. Change the number of questions students are allowed to ask (10 is still very doable)
2. Have students sketch a graph of h’’(x)
3. Make it competitive by having groups race against each other (we recommend not doing this right away because this activity is already very challenging for students)

Exam Insights

Outside of a slope field problem, students are rarely asked to create or sketch a function based on information about its derivatives. However, questions requiring graph analysis abound in the multiple choice sections. Students are often asked to identify a possible graph of f(x) or f’(x) when given information (tabular or graphical) for f’(x) or f”(x). Additionally, FRQs will focus on a student’s ability to communicate the candidates test, first derivative test, or second derivative test as justification of extrema.

Investigate the following items for excellent sample questions:
Past AB FRQs: 2018 AB 3, 2017 AB 3, 2016 AB 3, 2015 AB 5, 2014 AB 3, 2013 AB 4, 2012 AB 3

Past AB MCQs: 2018 AB #8, 16, 80, 88 and 2017 AB # 16, 27, 80, 86

Student Misconceptions

At this point in our study of derivatives and graph analysis, many vocabulary terms and new mathematical skills have been introduced. One can easily understand how the various terms (values vs. points, increasing vs. positive, etc.) and their graphical representations can be conflated. Use of correct and precise language by the teacher is required to help students discriminate between a positive value of the first derivative and the increasing behavior of a function --- or, the decreasing behavior of the first derivative and the negative value of the second derivative. Additionally, we must address --- both analytically and graphically --- the false assumption that a zero valued second derivative necessarily indicates a point of inflection. This belief persists in even the most talented students!

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