Sketching Graphs of Functions and Their Derivatives
Build a labeled calculus sketch from slope signs, concavity, domain behavior, and a function anchor.
College Board Topic 5.8 · CED p. 101 (PDF p. 106) · CED effective Fall 2020; project course outline 2026–27.
A calculus sketch is an argument about features, not a contest to draw a perfect curve. Gather the domain, intercepts or a known anchor, endpoint or limiting behavior, critical numbers, derivative signs, concavity, and inflection points. Label exact features when possible. A function formula supplies derivatives analytically; a derivative graph supplies signs and slopes; a table can supply exact values or carefully labeled estimates.
When sketching f′ from f, its height at x is the tangent slope of f. A horizontal tangent of f produces a derivative zero. A sharp corner of f generally creates an undefined derivative, so leave the relevant derivative point missing rather than drawing an invented smooth join. A maximum of f′ corresponds to a change in f’s concavity, not necessarily a maximum of f.
When sketching f from f′, use the derivative’s sign for increase/decrease and its slope for concavity. A known f-value fixes vertical placement. Without an anchor, adding a constant gives different f graphs with the same derivative. If numerical entries are only isolated samples, do not invent behavior between them unless the problem provides connecting information. A calculator plot can check your sketch, but the sign evidence should explain it even if the window is poor.
Let's work through it
Suppose f′(x) = x² − 1 and f(0) = 2. Derivative zeros are −1 and 1.
Why: These split the sign intervals and provide candidate extrema.
f increases outside [−1,1] and decreases between them.
Why: The derivative is positive outside and negative inside.
f″(x) = 2x, so f is concave down for x < 0 and up for x > 0.
Why: The derivative’s slope determines concavity.
Sketch a local maximum at x = −1, a local minimum at x = 1, and an inflection through the anchor (0,2).
Why: Combine monotonicity, sign changes, concavity, and the known function value; do not confuse f′(0) = −1 with f(0).
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Can f′ determine the vertical height of f without an anchor?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Can f′ determine the vertical height of f without an anchor?
Worked answer and reason
No; f and f + C share the same derivative.
Differentiation removes constant vertical shifts.
Retrieval 2
If f has a sharp corner, should a derivative sketch be smooth there?
Worked answer and reason
No; the derivative may be undefined there.
Left and right slopes need not match.
Catch the tempting mistake
- The derivative graph looks like a vertically shifted function graph.
- Its heights are slopes of f, so its shape can be very different.
- A sparse table determines a unique smooth graph.
- Many curves fit a finite table; keep unsupported details provisional.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 5 problem 17 · No calculator · analytic
- Unit 5 problem 18 · No calculator · graphical
- Unit 5 problem 29 · No calculator · multiple
For the teacher
- Grade feature labels and reasoning before artistic smoothness.
- Invite learners to give two different f sketches with the same f′ when no initial value is supplied.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
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Critical point
A critical number is an input in the function’s domain where its derivative is zero or does not exist. The corresponding point on the graph is called a critical point. For example, x³ has a critical number at 0, but it has no local maximum or minimum there. Critical numbers are candidates for extrema and need further analysis; inputs outside the domain are excluded.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.