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Connecting features of f, f’, and f’’ (Topics 5.8-5.9)

Unit 5 - Day 7

​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: Quiz, Quiz, Trade


Additional Materials:


Lesson Handout

Answer Key


Each student is given a card with a unique graph and a related question. Without help from others, the student answers the question on a sticky note. The sticky note is then attached to the back of the card. When directed, students walk around the room, find a partner, and show each other the fronts of their cards. The student must analyze the graph and answer the question on their partner’s card. The person holding the card then checks their partner’s answer against the sticky note. When both parties agree on the answers, students exchange cards and seek a different partner. This process repeats several times. If a student notices that an answer on the sticky note is incorrect, he/she should revise it. Have students carry around a pencil during this activity for these edits! At the end of the activity, students should retrieve their original card and see if corrections or changes have been made. Then have students complete the summary page.

Teaching Tips

Emphasize to students that this is a learning activity more so than a review activity. We expect students to make mistakes! It’s okay to have your answer revised by a peer.

We always enjoy playing this game along with our students. When a student is quizzing you, take the opportunity to model your thought process as you analyze the graph (e.g. “This is the derivative of f(x) which means the y-values represent slopes. If I want a relative minimum on f(x) I would need to see f’(x) changing from negative to positive which occurs at x=_____.”)

This activity allows students to work at their own pace, taking as much time as necessary to analyze a graph. It’s okay that some students will only make it through five cards while others go through 10 or 15. This would also make a great warm-up or review activity on subsequent days and can be used as a filler activity whenever you have an extra 5-10 minutes.

At the end of the activity, have students return to their seats and complete the Summary Page, which will solidify their understanding of the connections between the graphs of f, f’, and f’’.

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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