Worksheets · Calculus

Curve sketching worksheet

The two derivatives between them say everything the picture needs: the first says where the curve rises and where it turns over, and the second says which way it bends and where the bend changes.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Curve Sketching

Date Period

Sketch each curve or answer as asked, giving the extrema, inflection points and any asymptotes.

  1. Sketch the curve

    Sketch y=x3−3x2+3−10−10−5−5551010
  2. Sketch y=−23x3+4x2−6x+3−10−10−5−5551010
  3. Sketch y=−13x3+3x2−5x−2−10−10−5−5551010
  4. Sketch y=13x3−3x2+5x+3−10−10−5−5551010

  1. Find asymptotes, extrema and inflection points

    For y=xx2+9, give the asymptotes, extrema and inflection points.
  2. For y=x2x2+4, give the asymptotes, extrema and inflection points.
  3. For y=x2x2−9, give the asymptotes, extrema and inflection points.
  4. For y=x2x2+16, give the asymptotes, extrema and inflection points.

  1. Find relative extrema and inflection points

    For f(x)=2(x+2)5/3−10(x+2)2/3−5, find the relative extrema and the inflection points (x-values).
  2. For f(x)=3(x+3)5/3−15(x+3)2/3+1, find the relative extrema and the inflection points (x-values).
  3. For f(x)=2(x−2)4−4(x−2)2+1, find the relative extrema and the inflection points (x-values).
  4. For f(x)=x4−18x2−2, find the relative extrema and the inflection points (x-values).

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

Sigma Prep · sigmaprep.io/worksheets

Name

Curve SketchingAnswer keyVersion 1

Date Period

Sketch each curve or answer as asked, giving the extrema, inflection points and any asymptotes.

  1. Sketch the curve

    Sketch y=x3−3x2+3−10−5551010−10−5max at x=0, min at x=2, inflection at x=1
  2. Sketch y=−23x3+4x2−6x+3−10−10−5−5551010max at x=3, min at x=1, inflection at x=2
  3. Sketch y=−13x3+3x2−5x−2−10−10−5−5510510max at x=5, min at x=1, inflection at x=3
  4. Sketch y=13x3−3x2+5x+3−10−5551010−10−5max at x=1, min at x=5, inflection at x=3

  1. Find asymptotes, extrema and inflection points

    For y=xx2+9, give the asymptotes, extrema and inflection points.asymptote y=0; min (−3,−16), max (3,16); inflection at x=−33, 0, 33
  2. For y=x2x2+4, give the asymptotes, extrema and inflection points.asymptote y=1; min (0,0); inflection at x=−233, 233
  3. For y=x2x2−9, give the asymptotes, extrema and inflection points.asymptotes x=−3, x=3, y=1; max (0,0); no inflection points
  4. For y=x2x2+16, give the asymptotes, extrema and inflection points.asymptote y=1; min (0,0); inflection at x=−433, 433

  1. Find relative extrema and inflection points

    For f(x)=2(x+2)5/3−10(x+2)2/3−5, find the relative extrema and the inflection points (x-values).Max at x=−2Min at x=0Inflection at x=−3
  2. For f(x)=3(x+3)5/3−15(x+3)2/3+1, find the relative extrema and the inflection points (x-values).Max at x=−3Min at x=−1Inflection at x=−4
  3. For f(x)=2(x−2)4−4(x−2)2+1, find the relative extrema and the inflection points (x-values).Max at x=2Min at x=1 and x=3Inflection at x=2−33 and x=2+33
  4. For f(x)=x4−18x2−2, find the relative extrema and the inflection points (x-values).Max at x=0Min at x=−3 and x=3Inflection at x=−3 and x=3

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

How to do these

Sketch y=13x3−x2−3x

  1. The derivative is x squared minus 2x minus 3, which factors to x minus 3 times x plus 1.
  2. So the curve turns at x equals negative 1 and x equals 3. The second derivative is 2x minus 2, negative at negative 1 and positive at 3, so the first is a maximum and the second a minimum.
  3. The second derivative is zero at x equals 1, and it changes sign there, so that is the inflection point.

The answer is max at x=−1, min at x=3, inflection at x=1.

Where students go wrong

Sketching from the turning points alone. Two points and a wiggle can be drawn a dozen ways; the inflection point is what fixes which way the curve bends on the way between them.

Also called graphing using derivatives, graphing polynomials with calculus, sketching curves or first and second derivative graphing.

What you can put on this worksheet

Sketch the curve
Sketch y=−13x3+3x2−5x−1 → max at x=5, min at x=1, inflection at x=3
Relative maxima and minima
Find the relative extrema of y=6x2−8 → min (0,−8)
Increasing and decreasing
y=x3−3x2−45x−3. Where is y decreasing? → (−3,5)
Intervals of concavity
y=2x3+30x2+x. Where is y concave down? → (−∞,−5)
Rational functions
For y=x2x2+9, give the asymptotes, extrema and inflection points. → asymptote y=1; min (0,0); inflection at x=−3, 3
Extrema and inflection points together
For f(x)=2x4−4x2+4, find the relative extrema and the inflection points (x-values). → Max at x=0Min at x=−1 and x=1Inflection at x=−33 and x=33

Questions about these worksheets

Yes. Every curve sketching 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 curve sketching 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 6: sketch the curve, relative maxima and minima, increasing and decreasing, intervals of concavity, rational functions and extrema and inflection points together. Tick as many as you want and set how many of each, or let it spread them evenly.

Sketching from the turning points alone. Two points and a wiggle can be drawn a dozen ways; the inflection point is what fixes which way the curve bends on the way between them.