How do you build a graph from derivative evidence?
You will be able to: Combine domain, anchors, first-derivative signs and concavity in a consistent sketch.
BC foundation: Unit 5 shares these analytical differentiation objectives with AB. Check theorem hypotheses, justify derivative signs over intervals, and compare feasible candidates before claiming an optimum. State the local branch when analyzing an implicit relation.
How do you build a graph from derivative evidence?
A route description can say “climb, flatten, descend, then climb again.” Derivative information gives a similar set of instructions for sketching a function, but a starting height is needed to place it vertically.
A useful starting point: When does the second derivative classify a stationary point? →
Words and symbols before equations
- Anchor point
- A known coordinate fixing one graph height.
- Qualitative sketch
- A drawing showing justified behavior without inventing unsupported exact values.
- Intercept
- Where the graph meets a coordinate axis.
- Consistency check
- Confirming every feature agrees with the given evidence.
What this picture assumes
Original model; numerical readouts are rounded. f=x³−3x; f′=3x²−3; f″=6x. Algebra proves the interval sign chart. Separate plot axes have independently labeled scales. x and f are dimensionless.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- x=0.5: f=-1.375, f′=-2.25, f″=3. Negative slope; concave up at this input. C=0 affects only f’s heights. The plotted window is finite; exact interval signs come from the polynomial factors.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For f=x³−3x, the domain is all real numbers. Its zeros are −√3,0,√3, and its end behavior is downward to the left and upward to the right.
The first derivative 3x²−3 supplies increasing/decreasing intervals and stationary points (−1,2) and (1,−2). The second derivative 6x supplies concavity and inflection (0,0).
Sketch these features together: rise concave down to the left maximum, fall through the inflection, reach the right minimum, then rise concave up. A smooth drawing should satisfy all evidence, not only pass through marked points.
If only f′ and one height are given, exact other heights may require information beyond this unit. You can still sketch signs, turns and bending. Never infer extra oscillations or intercept locations from sparse data alone.
A worked example, step by step
Sketch a function with f′<0 for x<1, f′>0 for x>1, f″>0 everywhere and f(1)=2.
- The function decreases to x=1 and increases after it.
- It therefore has a minimum at the anchor (1,2), assuming continuity there.
- Positive second derivative makes the entire sketch concave up.
- A bowl such as (x−1)²+2 is one valid example, but the sign information does not uniquely force that formula.
A sketch represents the supplied evidence. It does not justify invented exact heights or a unique formula.
Does knowing f′ determine the vertical shift?
Compare with an explanation
No. Adding a constant to f changes its heights but not its derivatives.
Predict. Change one thing. Explain.
Read the function and derivative panels together. Identify each turning point and inflection before looking at the displayed function values.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x=0.5: f=-1.375, f′=-2.25, f″=3. Negative slope; concave up at this input. C=0 affects only f’s heights. The plotted window is finite; exact interval signs come from the polynomial factors.
| Interval | f′ sign | f behavior | f″ sign | Concavity |
|---|---|---|---|---|
| (−∞,−1) | + | increasing | − | down |
| (−1,0) | − | decreasing | − | down |
| (0,1) | − | decreasing | + | up |
| (1,∞) | + | increasing | + | up |
Original model; numerical readouts are rounded. f=x³−3x; f′=3x²−3; f″=6x. Algebra proves the interval sign chart. Separate plot axes have independently labeled scales. x and f are dimensionless.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the relevant derivative signs, theorem conditions, domain or geometric constraint. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionDescribe a consistent sketch for f=x³−3x+4, giving its local extrema and inflection, and explain its relation to x³−3x.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Adding 4 leaves f′=3x²−3 and f″=6x unchanged.
- 1 point: The local maximum is (−1,6).
- 1 point: The local minimum is (1,2), and inflection is (0,4).
- 1 point: The whole original cubic graph shifts upward 4 units with the same slopes and concavity.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What controls rising and falling?
First-derivative signs.
RECALL 2What controls bending?
How first derivatives change, or second-derivative signs.
RECALL 3What does a vertical shift preserve?
Both first and second derivatives.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you build a graph from derivative evidence?
- Audit domain → mark anchors → use f′ signs → use f″ signs → check consistency.
- A derivative graph does not fix vertical position without value information.
Remember: A sketch represents the supplied evidence. It does not justify invented exact heights or a unique formula.
Conditions: Original model; numerical readouts are rounded. f=x³−3x; f′=3x²−3; f″=6x. Algebra proves the interval sign chart. Separate plot axes have independently labeled scales. x and f are dimensionless.
Refresh Kid · AP Calculus BC Unit 5 · Objectives FUN-4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.8, FUN-4.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 5 has twelve official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Existence theorems require their stated hypotheses. Critical numbers must belong to the function’s domain; interior local extrema and included endpoints are handled explicitly. Extrema and inflection candidates need justification, not just a zero derivative. Sparse derivative samples do not establish interval-wide signs. Optimization includes the constraint, feasible domain and global comparison. Implicit-curve conclusions specify the branch and distinguish finite slopes from vertical tangents.
The Organic Chemistry Tutor Mean Value Theorem and Optimization Problems video titles, creator and relevant descriptions were checked; full videos were not reviewed. The free optimization video mentions additional paid material, which is not required here. Khan Academy’s unit destination was checked; its JavaScript lesson contents were not fully readable by the research tool. OpenStax Sections 4.3, 4.4, 4.5 and 4.7 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. The open-box model is original geometry using existing self-hosted Three.js with its MIT license. A 12 cm square sheet loses four equal corner squares. The remaining net folds into an open box; the geometry uses the same lengths as the labeled 2D net. Fold angle is a construction view, not an independent design variable; displayed volume refers to the fully upright box. Camera rotation changes no mathematical values. Complete 2D diagrams and explanations remain available without WebGL.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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