Florida B.E.S.T. MA.912.C.3.6
The Standard
Sketch graphs by using first and second derivatives. Compare the corresponding characteristics of the graphs of f, f' and f".
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use the sign and zeros of f' to locate increasing intervals, decreasing intervals, and possible extrema. They use f'' to determine concavity and possible inflection points. They connect these features across all three graphs.
What Mastery Looks Like
- Students can identify where f increases, decreases, reaches extrema, changes concavity, and has inflection points. They can sketch a consistent graph and explain how features of f, f', and f'' line up.
Common Misconceptions
- Students may think a zero of f' is an x-intercept of f rather than a possible maximum or minimum. They may confuse increasing with concave up. They may label every zero of f'' as an inflection point without checking for a concavity change.
How to Assess It
- Give students sign charts for f' and f''. Ask them to sketch one possible graph of f and label increasing intervals, extrema, concavity, and inflection points.
Lesson moves
Ways to Teach It
Sort card sets showing f, f', and f'' graphs, then justify each match with labeled zeros, extrema, and concavity changes.
Discuss: If f' is positive and decreasing, what must f be doing, and what sign should f'' have?
Play Graph Detective: reveal one derivative graph at a time, and award points for correctly predicting features of the other two graphs.
Use a position-time graph to connect velocity and acceleration with increasing, decreasing, concave up, and concave down behavior.
Free download
Printable MA.912.C.3.6 Worksheet
A ready-to-print activity worksheet aligned to MA.912.C.3.6, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetKeep exploring
Related Standards
- MA.912.C.4.5
Analyze function graphs by using derivative graphs and the Fundamental Theorem of Calculus.
- MA.912.C.2.6
Find derivatives of inverse functions.
- MA.912.C.2.7
Find second derivatives and derivatives of higher order.
- MA.912.F.1.7
Compare key features of two functions each represented algebraically, graphically, in tables or written descriptions.
Turn this exact standard into a lesson
Grade, subject, topic, and the complete standard are prefilled. Create one free, no account needed.