Calculate Average Change across an Interval

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A function has f(2)=2 and f(4)=8. What is its average rate of change from x=2 to x=4? Candidate answers: (A) 2 (B) 3 (C) 4 (D) 6.
• Independent check inputs: A function has f(2)=3 and f(4)=12. What is its average rate of change from x=2 to x=4? Candidate answers: (A) 3.5 (B) 5.5 (C) 9 (D) 9/2.

Learn the idea

A function assigns one output to each allowed input. A table can reveal nonlinear structure or interval-average change, while the domain records which inputs make sense mathematically and in the supplied context. Average rate of change measures the net output change per net input change between two endpoints. It is the slope of the secant connecting those endpoint points.

Here is the method on the supplied teaching case. Use change in output divided by change in input: (8−2)/(4−2)=6/2=3. This is the interval average; it need not hold at every point.

Keep subtraction order consistent in numerator and denominator. An interval average does not prove that the function changes at that rate at every intermediate input, particularly for a nonlinear function.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A function has f(2)=9 and f(4)=36. What is its average rate of change from x=2 to x=4? Candidate answers: (A) 12.5 (B) 14.5 (C) 27 (D) 27/2. Show the mathematical reason, not only the selected answer. Expected result: 27/2. Use change in output divided by change in input: (36−9)/(4−2)=27/2=27/2. This is the interval average; it need not hold at every point.

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