Mississippi Math C.G.23

MathGrades 9–12Geometry (G) Geometry

The Standard

Apply differentiation techniques to curve sketching

2025 Mississippi College- and Career-Readiness Standards for Mathematics · Employ Various Integration Properties and Techniques to Elevate Integrals

Teacher's field guide

What Students Need to Do: Students find where a curve rises, falls, or has a horizontal tangent by checking its derivative. They use the second derivative to check how the curve bends. Then they use those clues to sketch the graph.
What Mastery Looks Like: A student finds critical points and checks whether the derivative changes sign at each one. Their sketch shows the correct peaks, valleys, and bending, with intercepts placed accurately.
Common Misconceptions: Students may call every point where the derivative is zero a peak or valley. They may also confuse a negative derivative, which means the curve is falling, with a negative second derivative, which means it bends downward.
How to Assess It: Give students f(x) = x³ − 3x on −2 ≤ x ≤ 2. Ask them to find where it rises and falls, label its turning points, and sketch the curve.

Lesson moves

Give pairs a printed graph of f'(x) = 3x² − 3 and f(0) = 0; have them mark sign changes and sketch f.
Ask students to explain why f'(2) = 0 does not guarantee a peak, using f(x) = (x − 2)³.
Have pairs match derivative sign-chart cards to curve-sketch cards, then explain one match to another pair.
Use h(t) = −5t² + 20t + 1 for a tossed ball; find its peak time and sketch its height from 0 to 4 seconds.

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