Worksheets · Algebra 2

Average rate of change worksheet for Algebra 2

Algebra 2 applies average rate of change to polynomials, simple rational functions like 9/x and tables of data, and asks whether the rates over equal steps are increasing or decreasing. You also simplify the difference quotient for a quadratic, which previews the slope ideas of calculus.

Also taught in Precalculus, Calculus and Algebra 1, each at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Average Rate of Change

Date Period

Find each rate of change, and answer what is asked. Leave answers exact.

  1. Find the average rate of change

    Average rate of change of f(x)=−3x3−7x+5 on [0,4]
  2. Average rate of change of f(x)=x2−8x on [5,10]
  3. Average rate of change of f(x)=−9x on [5,9]
  4. Average rate of change of f(x)=−5x−6 on [−7,−3]
  5. Average rate of change of f(x)=9x on [−4,−2]

  1. Simplify f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}

    Simplify f(x+h)−f(x)h for f(x)=−3x2−6x−8
  2. Simplify f(x+h)−f(x)h for f(x)=3x2+2x−8
  3. Simplify f(x+h)−f(x)h for f(x)=−x2+4x+5
  4. Simplify f(x+h)−f(x)h for f(x)=5x2+4x+6
  5. Simplify f(x+h)−f(x)h for f(x)=x2−6x−2

  1. Find average rates of change from tables

    Find the average rate of change from x=1 to x=10. x136910f(x)−6146227
  2. Find the average rate of change from x=4 to x=8. x−11468f(x)9192313−2
  3. Find the average rate of change from x=3 to x=9. x036911f(x)2821−41128
  4. Find the average rate of change from x=−3 to x=−1. x−3−2−114f(x)17−428−1629
  5. Find the average rate of change from x=3 to x=8. x23568f(x)−1930−5−16−20

  1. Compare average rates over intervals

    Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)68163050
  2. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)20−4−12−28
  3. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)−5−13711
  4. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)−1−3−5−7−9
  5. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)4−1−10−23−40

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

Sigma Prep · sigmaprep.io/worksheets

Name

Average Rate of ChangeAnswer keyVersion 3

Date Period

Find each rate of change, and answer what is asked. Leave answers exact.

  1. Find the average rate of change

    Average rate of change of f(x)=−3x3−7x+5 on [0,4]−55
  2. Average rate of change of f(x)=x2−8x on [5,10]7
  3. Average rate of change of f(x)=−9x on [5,9]15
  4. Average rate of change of f(x)=−5x−6 on [−7,−3]−5
  5. Average rate of change of f(x)=9x on [−4,−2]−98

  1. Simplify f(x+h)−f(x)h\frac{f(x + h) - f(x)}{h}

    Simplify f(x+h)−f(x)h for f(x)=−3x2−6x−8−6x−3h−6
  2. Simplify f(x+h)−f(x)h for f(x)=3x2+2x−86x+3h+2
  3. Simplify f(x+h)−f(x)h for f(x)=−x2+4x+5−2x−h+4
  4. Simplify f(x+h)−f(x)h for f(x)=5x2+4x+610x+5h+4
  5. Simplify f(x+h)−f(x)h for f(x)=x2−6x−22x+h−6

  1. Find average rates of change from tables

    Find the average rate of change from x=1 to x=10. x136910f(x)−6146227139
  2. Find the average rate of change from x=4 to x=8. x−11468f(x)9192313−2−254
  3. Find the average rate of change from x=3 to x=9. x036911f(x)2821−41128−53
  4. Find the average rate of change from x=−3 to x=−1. x−3−2−114f(x)17−428−1629112
  5. Find the average rate of change from x=3 to x=8. x23568f(x)−1930−5−16−20−10

  1. Compare average rates over intervals

    Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)681630502, 8, 14, 20; increasing, so the graph is concave up
  2. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)20−4−12−28−2, −4, −8, −16; decreasing, so the graph is concave down
  3. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)−5−137114, 4, 4, 4; the same, so the graph is a line
  4. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)−1−3−5−7−9−2, −2, −2, −2; the same, so the graph is a line
  5. Find the average rate of change over each interval. Are the rates increasing,decreasing or the same, and what does that show?x01234f(x)4−1−10−23−40−5, −9, −13, −17; decreasing, so the graph is concave down

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

How to do these

Simplify f(x+h)−f(x)h for f(x)=x2+3x

  1. f(x + h) = x² + 2xh + h² + 3x + 3h.
  2. Subtract f(x): 2xh + h² + 3h.
  3. Divide every term by h.

The answer is 2x+h+3.

Where students go wrong

Squaring x + h as x² + h². (x + h)² is x² + 2xh + h², and without the 2xh the difference quotient loses its x term.

Also called average rate of change over an interval, difference quotient or slope of a secant line.

What you can put on this worksheet

Average rate of change over an interval
Average rate of change of f(x)=3x2−7x on [1,2] → 2
The difference quotient
Simplify f(0+h)−f(0)h for f(x)=2x2+3x−8 → 2h+3
The equation of the secant line
Write the equation of the secant line of f(x)=2x2+3x−8 from x=−1 to x=2 → y=5x−4
The general difference quotient
Simplify f(x+h)−f(x)h for f(x)=4x2−8x → 8x+4h−8
From a table
Find the average rate of change from x=5 to x=6. x0256f(x)8−829−3 → −32
Are the rates increasing or decreasing?
Find the average rate of change over each interval. Are the rates increasing,
decreasing or the same, and what does that show?
x01234f(x)2−3−8−13−18 → −5, −5, −5, −5; the same, so the graph is a line
Average speed and the speed limit
A car passes mile marker 32, and 30 minutes later it passes mile marker 67. The speed limit is 55 mph. Must it have gone faster than the limit at some moment? → Average speed 70 mph; yes: by the mean value theorem it went exactly 70 mph at some moment
From a graph, fractional endpoints
The graph of f(x)=12x2+x+2 is shown. Find its average rate of change on [−72,−1] → −54

Questions about these worksheets

Yes. Every average rate of change 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.

Algebra 2 is usually taken around 10th or 11th grade, 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 average rate of change 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 8: average rate of change over an interval, the difference quotient, the equation of the secant line, the general difference quotient, from a table, are the rates increasing or decreasing?, average speed and the speed limit and from a graph, fractional endpoints. Tick as many as you want and set how many of each, or let it spread them evenly.

Dividing by the wrong run, or subtracting the outputs in one order and the inputs in the other. Keep the same endpoint on top and bottom: f(b) minus f(a) over b minus a, in that order both times.