Worksheets · Calculus

Analyzing the graph of f' worksheet

The graph shown is not f. Above the axis means f is rising; a crossing from above to below is a high point of f. Where f' itself rises, f is concave up, and where f' turns, f has an inflection point. Areas under f' are changes in f.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Reading f From the Graph of f′

Date Period

Use the graph of f′ to answer each question.

  1. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative minimum?−10−10−5−500551010
  2. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−10−10−5−500551010
  3. The graph of f′, the derivative of f, is shown. If f(1)=4, find f(4).−2−10123456−3−2−10123f′
  4. The graph of f′, the derivative of f, is shown. On what intervals is f increasing?−10−10−5−500551010
  5. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010
  6. The graph of f′, the derivative of f, is shown. If f(−3)=−1, find f(3).−3−2−101234−3−2−10123f′
  7. The graph of f′, the derivative of f, is shown. On what intervals is f decreasing?−10−10−5−500551010
  8. The graph of f′, the derivative of f, is shown. On what intervals is f concave up?−10−10−5−500551010
  9. The graph of f′, the derivative of f, is shown. If f(−3)=7, find f(0).−3−2−10123−3−2−10123f′
  10. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative maximum?−10−10−5−500551010
  11. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−10−10−5−500551010
  12. The graph of f′, the derivative of f, is shown. If f(1)=−5, find f(3).−4−3−2−10123−3−2−10123f′
  13. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative minimum?−10−10−5−500551010
  14. The graph of f′, the derivative of f, is shown. On what intervals is f concave up?−10−10−5−500551010
  15. The graph of f′, the derivative of f, is shown. If f(−3)=−1, find f(−1).−3−2−1012345−3−2−10123f′
  16. The graph of f′, the derivative of f, is shown. On what intervals is f increasing?−10−10−5−500551010
  17. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010
  18. The graph of f′, the derivative of f, is shown. If f(−2)=8, find f(−1).−3−2−10123−3−2−10123f′
  19. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative maximum?−10−10−5−500551010
  20. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Reading f From the Graph of f′Answer keyVersion 1

Date Period

Use the graph of f′ to answer each question.

  1. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative minimum?−10−10−5−500551010x=−1
  2. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−10−10−5−500551010x=−4 and x=−1
  3. The graph of f′, the derivative of f, is shown. If f(1)=4, find f(4).−2−10123456−3−2−10123f′32
  4. The graph of f′, the derivative of f, is shown. On what intervals is f increasing?−10−10−5−500551010(0,3)∪(7,∞)
  5. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010(−1,3)
  6. The graph of f′, the derivative of f, is shown. If f(−3)=−1, find f(3).−3−2−101234−3−2−10123f′−6
  7. The graph of f′, the derivative of f, is shown. On what intervals is f decreasing?−10−10−5−500551010(−8,−6)∪(−1,∞)
  8. The graph of f′, the derivative of f, is shown. On what intervals is f concave up?−10−10−5−500551010(−∞,0)∪(2,∞)
  9. The graph of f′, the derivative of f, is shown. If f(−3)=7, find f(0).−3−2−10123−3−2−10123f′4
  10. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative maximum?−10−10−5−500551010x=−2
  11. The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection?−10−10−5−500551010x=−1 and x=1
  12. The graph of f′, the derivative of f, is shown. If f(1)=−5, find f(3).−4−3−2−10123−3−2−10123f′−7
  13. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative minimum?−10−10−5−500551010x=3
  14. The graph of f′, the derivative of f, is shown. On what intervals is f concave up?−10−10−5−500551010(−2,0)
  15. The graph of f′, the derivative of f, is shown. If f(−3)=−1, find f(−1).−3−2−1012345−3−2−10123f′2
  16. The graph of f′, the derivative of f, is shown. On what intervals is f increasing?−10−10−5−500551010(0,3)∪(8,∞)
  17. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010(−∞,−3)∪(0,∞)
  18. The graph of f′, the derivative of f, is shown. If f(−2)=8, find f(−1).−3−2−10123−3−2−10123f′8
  19. The graph of f′, the derivative of f, is shown. At what x-values does f have a relative maximum?−10−10−5−500551010x=−7 and x=0
  20. The graph of f′, the derivative of f, is shown. On what intervals is f concave down?−10−10−5−500551010(−∞,−6)∪(−2,∞)

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

The graph of f′ crosses the axis from above to below at x=2. What happens to f there?

  1. Left of 2, f' > 0, so f is increasing.
  2. Right of 2, f' < 0, so f is decreasing.
  3. Rising then falling: f has a relative maximum at x = 2.

The answer is A relative maximum at x=2.

Where students go wrong

Reading the graph as f. The high point of the graph of f' is an inflection point of f, not a maximum of f.

Also called graph of the derivative, f' graph, reading a derivative graph or f from f'.

What you can put on this worksheet

Increasing, decreasing and extrema from a curved f'
The graph of f′, the derivative of f, is shown. On what intervals is f increasing? → (−∞,−1)∪(5,∞)
Concavity and inflection from a curved f'
The graph of f′, the derivative of f, is shown. On what intervals is f concave up? → (−∞,0)
Values of f from the area under f'
The graph of f′, the derivative of f, is shown. If f(−1)=6, find f(4). — the graph of f′ on [-2, 5] → 352
Read f from the graph of f'
The graph of f′, the derivative of f, is shown. At what x does f have a relative minimum on (−1,8)? — the graph of f′ on [-1, 8] → x=2
Concavity from the graph of f'
The graph of f′, the derivative of f, is shown. At what x does f have a point of inflection? — the graph of f′ on [-1, 8] → x=5

Questions about these worksheets

Yes. Every analyzing the graph of f' sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the analyzing the graph of f' questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: increasing, decreasing and extrema from a curved f', concavity and inflection from a curved f', values of f from the area under f', read f from the graph of f' and concavity from the graph of f'. Tick as many as you want and set how many of each, or let it spread them evenly.

Reading the graph as f. The high point of the graph of f' is an inflection point of f, not a maximum of f.