Rates of Change in Functions

The rate of change of a function describes how quickly its output values change relative to changes in its input. For any function f(x), we can measure this change over a specific interval [a, b] by computing the average rate of change, which gives us a single number summarising the overall behaviour of the function between two points. This concept extends the familiar idea of slope from linear functions to all types of functions, including polynomials of any degree. Unlike linear functions, whose rate of change is constant everywhere, polynomial functions of degree two or higher have rates of change that vary across different intervals. A quadratic function might be decreasing rapidly on one interval and increasing slowly on another. By computing average rates of change over smaller and smaller intervals, we can begin to understand the instantaneous behaviour of a function at a single point — a concept that bridges algebra and calculus. Rates of change appear throughout science and everyday life: the speed of a falling object, the growth rate of a population, the marginal cost in economics. Mastering interval-based rates of change for polynomial functions builds the quantitative reasoning needed for advanced mathematics and provides the foundation for understanding derivatives in calculus.

Key takeaways
  • Define the interval and function values
  • Apply the average rate of change formula
  • Interpret the sign and magnitude
  • Shrink the interval to approach instantaneous rate
  • Compare rates across different intervals

Define the interval and function values

Choose two x-values, a and b, with a < b, forming the closed interval [a, b]. Evaluate the function at both endpoints to obtain f(a) and f(b). These two points, (a, f(a)) and (b, f(b)), lie on the graph of the function and define a secant line through the curve.

Apply the average rate of change formula

The average rate of change (AROC) is the slope of the secant line connecting the two points. It equals the change in output divided by the change in input: AROC = [f(b) − f(a)] / (b − a). This ratio measures how much f(x) changes, on average, for each one-unit increase in x over the interval.

Interpret the sign and magnitude

A positive AROC means the function is, on average, increasing over [a, b]. A negative AROC means it is, on average, decreasing. A larger absolute value indicates a steeper average rise or fall. For a linear function f(x) = mx + c, the AROC equals m for every interval, since the graph is a straight line. For a polynomial of degree ≥ 2, the AROC depends on which interval you choose.

Shrink the interval to approach instantaneous rate

If we fix a and let b approach a (written b → a), the secant line rotates toward the tangent line at x = a. The limiting value of [f(b) − f(a)] / (b − a) as b → a is called the instantaneous rate of change, or the derivative f′(a). While calculating this limit rigorously belongs to calculus, experimenting numerically by taking b = a + 0.1, a + 0.01, a + 0.001 shows students how the AROC stabilises toward a single value.

Compare rates across different intervals

Computing the AROC on multiple sub-intervals of a larger domain reveals where a polynomial is growing fastest or slowest. For a cubic f(x) = x³ − 3x, the function decreases on one interval and increases on another, and the rates of change on those intervals will have opposite signs. Tabulating AROCs systematically helps sketch accurate graphs and identify turning points.

The Key Equation

AROC=f(b)−f(a)b−a\text{AROC} = \frac{f(b) - f(a)}{b - a}
  • f(b) — Output at b (units of f): The value of the function evaluated at the right endpoint of the interval.
  • f(a) — Output at a (units of f): The value of the function evaluated at the left endpoint of the interval.
  • b - a — Change in input (units of x): The length of the interval; must be non-zero (b ≠ a).
  • \text{AROC} — Average rate of change (units of f per unit of x): The slope of the secant line through the points (a, f(a)) and (b, f(b)).

Worked Example

Problem. Let f(x) = x³ − 4x. (a) Find the average rate of change of f on the interval [1, 3]. (b) Find the average rate of change on [1, 1.5] and on [1, 1.1]. (c) Comment on what these results suggest about the instantaneous rate of change at x = 1.

AROC[1,3]=9,AROC[1,1.5]=0.75,AROC[1,1.1]=−0.69  ⟶  f′(1)=−1\text{AROC}_{[1,3]} = 9,\quad \text{AROC}_{[1,1.5]} = 0.75,\quad \text{AROC}_{[1,1.1]} = -0.69 \;\longrightarrow\; f'(1) = -1

Answer. (a) AROC on [1,3] = 9; (b) AROC on [1,1.5] = 0.75 and AROC on [1,1.1] = −0.69; (c) the values approach −1, which is the instantaneous rate of change (derivative) at x = 1.

Figure
Diagram
Secant Line vs Tangent Line on f(x) = x³ − 4xxy(a, f(a))(b, f(b))SecantTangent at af(x)=x³−4xThe red secant line gives the AROC between two points. The green tangent shows the instantaneous rate.
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton developed the method of fluxions — now called differential calculus — around 1666, formalising the concept of instantaneous rate of change. He recognised that the limiting value of the average rate of change (the slope of the secant as the interval shrinks to zero) gave the 'fluxion' of a quantity, directly linking average and instantaneous rates of change for polynomial and other functions. Newton reportedly conceived several of his greatest ideas, including early work on calculus and gravitation, during 1665–1666 when Cambridge University closed due to the bubonic plague and he retreated to his family farm in Woolsthorpe.

  • Change in Tandem — Change in Tandem explores how two quantities change simultaneously, directly extending the idea of rate of change from a single interval to covarying relationships between variables.
  • Polynomial End Behaviour — End behaviour describes what happens to a polynomial's output as x grows very large or very small — precisely where rates of change become extreme — linking overall function growth to rate concepts.
  • Polynomial & Rational Modeling — Real-world models using polynomials or rational functions are often interpreted through their rates of change (e.g. speed, growth, cost), making this concept central to applying models meaningfully.
  • Polynomial Symmetry — Symmetry of even and odd polynomial functions implies predictable sign patterns in their rates of change across symmetric intervals, reinforcing how function structure constrains rate behaviour.
  • Vertical and Horizontal Transformations — Scaling and shifting a function directly alters its rates of change: a vertical stretch by factor k multiplies the AROC by k, while a horizontal stretch by k divides it by k.
  • Equivalent Algebraic Forms — Rewriting a polynomial in factored or expanded form can simplify the algebra when computing f(b)−f(a), especially for higher-degree polynomials.
  • Binomial Expansion — The Binomial Theorem is used to expand f(a+h) for polynomial functions, which is the algebraic starting point for computing (f(a+h)−f(a))/h and ultimately finding derivative formulas.
  • Rational Function Zeros — Zeros of a rational function are points where the output is zero; rates of change at and near these zeros help characterise how the function crosses or touches the x-axis.
  • Rational Function Asymptotes — Near vertical asymptotes, the rate of change of a rational function grows without bound, making rate of change analysis essential for understanding asymptotic behaviour.
  • Holes in Rational Graphs — Holes occur where a rational function is undefined at a single point, raising questions about whether a meaningful rate of change can be defined nearby — connecting to limit-based thinking.