Worksheets · Calculus

Increasing and decreasing intervals worksheet

A positive derivative means rising and a negative one means falling, so the sign chart of the derivative answers the whole question.

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Name

Increasing and Decreasing Intervals

Date Period

Answer each question about the curve.

  1. Find where yy is increasing or decreasing

    y=x3+6x2−15x+1. Where is y decreasing?
  2. y=x3−6x2−1. Where is y decreasing?
  3. y=x3−75x−2. Where is y increasing?
  4. y=x3−9x2+24x−8. Where is y increasing?
  5. y=x3+3x2−45x+4. Where is y increasing?

  1. Use the graph of f′f' to find where ff increases, decreases or has extrema

    The graph of f′, the derivative of f, is shown. On what open intervals is f increasing?−10123456−3−2−10123f′
  2. The graph of f′, the derivative of f, is shown. On what open intervals is f decreasing?−10123456−3−2−10123f′
  3. The graph of f′, the derivative of f, is shown. At what x does f have a relative maximum on (−1,7)?−101234567−3−2−10123f′
  4. The graph of f′, the derivative of f, is shown. At what x does f have a relative minimum on (−3,5)?−3−2−1012345−3−2−10123f′
  5. The graph of f′, the derivative of f, is shown. At what x does f have a relative maximum on (−3,5)?−3−2−1012345−3−2−10123f′

  1. Find where non-polynomial functions increase or decrease

    Where is y=(x+1)ex increasing?
  2. Where is y=(x−2)ex decreasing?
  3. Where is y=x−5ln⁡x decreasing?
  4. Where is y=x−7ln⁡x increasing?
  5. Where is y=xex increasing?

  1. Find where ff increases or decreases across asymptotes and cusps

    Find the intervals on which f(x)=2x2x+3 is increasing and decreasing.
  2. Find the intervals on which f(x)=4 ⁣(x2−5)x−3 is increasing and decreasing.
  3. Find the intervals on which f(x)=4 ⁣(x2+3)x−1 is increasing and decreasing.
  4. Find the intervals on which f(x)=x2−8x+3 is increasing and decreasing.
  5. Find the intervals on which f(x)=4x2x−4 is increasing and decreasing.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Increasing and Decreasing IntervalsAnswer keyVersion 1

Date Period

Answer each question about the curve.

  1. Find where yy is increasing or decreasing

    y=x3+6x2−15x+1. Where is y decreasing?(−5,1)
  2. y=x3−6x2−1. Where is y decreasing?(0,4)
  3. y=x3−75x−2. Where is y increasing?(−∞,−5)∪(5,∞)
  4. y=x3−9x2+24x−8. Where is y increasing?(−∞,2)∪(4,∞)
  5. y=x3+3x2−45x+4. Where is y increasing?(−∞,−5)∪(3,∞)

  1. Use the graph of f′f' to find where ff increases, decreases or has extrema

    The graph of f′, the derivative of f, is shown. On what open intervals is f increasing?−10123456−3−2−10123f′(−1,1)
  2. The graph of f′, the derivative of f, is shown. On what open intervals is f decreasing?−10123456−3−2−10123f′(0,2)
  3. The graph of f′, the derivative of f, is shown. At what x does f have a relative maximum on (−1,7)?−101234567−3−2−10123f′x=2
  4. The graph of f′, the derivative of f, is shown. At what x does f have a relative minimum on (−3,5)?−3−2−1012345−3−2−10123f′x=−1
  5. The graph of f′, the derivative of f, is shown. At what x does f have a relative maximum on (−3,5)?−3−2−1012345−3−2−10123f′x=4

  1. Find where non-polynomial functions increase or decrease

    Where is y=(x+1)ex increasing?(−2,∞)
  2. Where is y=(x−2)ex decreasing?(−∞,1)
  3. Where is y=x−5ln⁡x decreasing?(0,5)
  4. Where is y=x−7ln⁡x increasing?(7,∞)
  5. Where is y=xex increasing?(−1,∞)

  1. Find where ff increases or decreases across asymptotes and cusps

    Find the intervals on which f(x)=2x2x+3 is increasing and decreasing.Increasing: (−∞,−6)∪(0,∞)Decreasing: (−6,−3)∪(−3,0)
  2. Find the intervals on which f(x)=4 ⁣(x2−5)x−3 is increasing and decreasing.Increasing: (−∞,1)∪(5,∞)Decreasing: (1,3)∪(3,5)
  3. Find the intervals on which f(x)=4 ⁣(x2+3)x−1 is increasing and decreasing.Increasing: (−∞,−1)∪(3,∞)Decreasing: (−1,1)∪(1,3)
  4. Find the intervals on which f(x)=x2−8x+3 is increasing and decreasing.Increasing: (−∞,−4)∪(−2,∞)Decreasing: (−4,−3)∪(−3,−2)
  5. Find the intervals on which f(x)=4x2x−4 is increasing and decreasing.Increasing: (−∞,0)∪(8,∞)Decreasing: (0,4)∪(4,8)

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

How to do these

Where is y=x3−12x increasing?

  1. The derivative is 3x squared minus 12.
  2. That is zero at x equals 2 and minus 2.
  3. It is positive outside those.

The answer is (−∞,−2)∪(2,∞).

Where students go wrong

Answering with y values. Intervals of increase are intervals of x, because the question is about where along the axis the curve is rising, not how high it gets.

Also called intervals of increase, intervals of decrease, monotonic intervals or sign chart of the derivative.

What you can put on this worksheet

Increasing and decreasing
y=x3−3x2−45x−3. Where is y decreasing? → (−3,5)
Critical points
Find the critical points of y=2x2−8 → x=0
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
Beyond polynomials
Where is y=x−5ln⁡x decreasing? → (0,5)
Across an asymptote and through a cusp
Find the intervals on which f(x)=3 ⁣(x2−8)x−3 is increasing and decreasing. → Increasing: (−∞,2)∪(4,∞)Decreasing: (2,3)∪(3,4)

Questions about these worksheets

Yes. Every increasing and decreasing intervals 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 increasing and decreasing intervals 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 two 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 and decreasing, critical points, read f from the graph of f', beyond polynomials and across an asymptote and through a cusp. Tick as many as you want and set how many of each, or let it spread them evenly.

Answering with y values. Intervals of increase are intervals of x, because the question is about where along the axis the curve is rising, not how high it gets.