Unit 1: Polynomial & Rational Functions

Rates of Change

Relating average and instantaneous rates of change to a function's graph.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

What a Changing Rate of Change Reveals

A linear function has exactly one rate of change, the same everywhere: its slope. Most functions aren't linear, though, and their rate of change shifts from one part of the graph to another. Before reading on, consider a function whose rate of change is itself changing — is there still a pattern hiding in how it changes, or is it just noise?

Definition — Average Rate of Change

The average rate of change of f on the interval [a, b] is (f(b) − f(a)) / (b − a) — the slope of the secant line connecting the two points (a, f(a)) and (b, f(b)). It measures how much the output changed per unit of input, on average, across the whole interval.

\text{AROC on } [a,b] = \frac{f(b) - f(a)}{b - a}

For a linear function, the average rate of change is the same number no matter which interval you pick — that's exactly what 'constant rate of change' means. For a curved graph it isn't constant, but the way it changes still says something concrete about the graph's shape: if the average rate of change is increasing across consecutive intervals, the graph is bending upward (concave up); if it's decreasing, the graph is bending downward (concave down).

Worked Example — Computing an Average Rate of Change

Find the average rate of change of f(x) = x² − x on [1, 3]. f(3) = 9 − 3 = 6 and f(1) = 1 − 1 = 0, so the average rate of change is (6 − 0)/(3 − 1) = 3. On average, f's output rose 3 units for every 1-unit increase in x across that interval.

x	f(x) = x² − x	1st difference	2nd difference
0	0	—	—
1	0	0	—
2	2	2	2
3	6	4	2
4	12	6	2

Equal-width input steps (Δx = 1) for f(x) = x² − x.

The 1st-difference column is exactly the average rate of change over each one-unit interval, and it isn't constant — it climbs steadily: 0, 2, 4, 6. But look at how it climbs: by exactly 2 every time. Differencing that already-changing rate of change produces something constant. That's not a coincidence of this particular function.

Definition — Constant Differences and Polynomial Degree

For a degree-n polynomial, sampled at equally spaced input values, the n-th differences of the outputs are constant. A degree-1 (linear) function has constant 1st differences — that's just its constant slope, restated as a table. A degree-2 (quadratic) function's rate of change is itself changing like a linear sequence, so its 2nd differences are constant. Each additional degree pushes the 'constant' pattern one difference level deeper.

Worked Example — Identifying a Cubic from a Difference Table

Confirm that f(x) = x³ is degree 3 using a difference table at x = 0, 1, 2, 3, 4: outputs 0, 1, 8, 27, 64. First differences: 1, 7, 19, 37. Second differences: 6, 12, 18. Third differences: 6, 6 — constant. Reaching a constant row at the third difference confirms the data comes from a degree-3 polynomial, without ever seeing the equation itself.

Tip

When a difference table doesn't look constant yet, keep differencing one level deeper before concluding the data isn't polynomial — a degree-4 function's first three difference rows are all still changing.

Common Mistakes

Computing an average rate of change as f(b) − f(a) alone, forgetting to divide by the interval width (b − a).
Average rate of change is a slope, a ratio of output-change to input-change — always divide by (b − a), even when the interval happens to have width 1 and the division looks unnecessary.
Expecting the 1st-difference row itself to be constant for any non-linear function.
Only a linear function has constant 1st differences. A quadratic's 1st differences form their own increasing (or decreasing) pattern — it's the 2nd differences that come out constant.

Key Takeaways

Average rate of change on [a, b] is (f(b) − f(a))/(b − a), the slope of the secant line through the endpoints.
Whether the average rate of change increases or decreases across consecutive intervals reveals whether the graph is concave up or concave down.
For a degree-n polynomial sampled at equally spaced inputs, the n-th differences of the outputs are constant — a table alone can reveal a polynomial's degree.

Summary

Average rate of change turns a curve's changing steepness into a number you can compute and compare, and constant differences turn that idea into a way to detect a polynomial's degree directly from a table. The next lesson uses degree — together with the leading coefficient — to predict what a polynomial's graph does infinitely far to the left and right, and how many zeros it can have.

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