Average Rate of Change

Statement

The average rate of change of a function between two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) is given by:

\[\text{Average rate on } [x_1, x_2] = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]

This formula represents the slope of the secant line through the two points on the function.

Why it’s true (short reason)

• The change in function values is \(f(x_2) - f(x_1)\).
• The change in input values is \(x_2 - x_1\).
• Dividing change in output by change in input gives the rate of change (rise over run).
• This is the same principle as slope in coordinate geometry.

Recipe (how to use it)

1. Identify the two \(x\)-values: \(x_1\) and \(x_2\).
2. Find their corresponding \(y\)-values by calculating \(f(x_1)\) and \(f(x_2)\).
3. Subtract: \(f(x_2) - f(x_1)\).
4. Subtract: \(x_2 - x_1\).
5. Divide the differences to find the average rate of change.

Spotting it

You use this formula when:

• The question asks for “average rate of change” between two points.
• You are working with functions, graphs, or tables of values.
• You want the slope of a line through two points on a curve.

Common pairings

• Linear functions, where average rate of change equals the constant slope.
• Differentiation, where the instantaneous rate of change is the limit of this formula.
• Physics problems involving average speed or velocity.

Mini examples

1. Given: \(f(x) = x^2\). Find average rate of change between \(x=2\) and \(x=5\). \(f(2)=4, f(5)=25\). \(\frac{25-4}{5-2} = \frac{21}{3} = 7\).
2. Given: \(f(x) = 3x+1\). Find average rate of change between \(x=1\) and \(x=4\). \(f(1)=4, f(4)=13\). \(\frac{13-4}{4-1} = \frac{9}{3} = 3\).

Pitfalls

• Forgetting to divide by \(x_2 - x_1\).
• Switching the order of subtraction incorrectly (which can flip the sign).
• Using instantaneous rate instead of average rate.
• Confusing it with arithmetic average of function values.

Exam strategy

• Write both function values clearly before subtraction.
• Check that denominator \(x_2-x_1\) is non-zero.
• Use exact fractions unless decimals are required.
• Interpret the result in context (e.g. speed, growth rate).

Summary

The average rate of change formula measures how quickly a function changes between two points. It is essentially the slope of the secant line. It is the foundation for understanding gradients, linear models, and the concept of derivatives in calculus.