For a linear function, the average rate of change over any length input-value interval is constant. To see why, take f(x) = mx + b and any interval [p, q]. The output changes by (mq + b) − (mp + b) = m(q − p), and the input changes by q − p, so the ratio is exactly m. Short interval, long interval, left side, right side: always m.
Build a linear function f(x) = mx + b, then pick any interval you like. Short or long, left or right, the average rate of change comes out the same.
| other interval | length | average rate of change |
|---|---|---|
| [-4, -3] | 1 | 2.00 |
| [0, 1] | 1 | 2.00 |
| [-3, 4] | 7 | 2.00 |
| [2, 5] | 3 | 2.00 |
A quadratic function f(x) = ax² + bx + c does not have one rate. The average rate of change depends on where the interval sits. But there is a pattern. For consecutive equal-length input-value intervals, the average rates of change can be given by a linear function. If each interval has length h, then
The right side is a linear function of the interval's midpoint x + h/2, with slope 2a. Slide a and b and watch the points on the right-hand graph: they always fall on a straight line.
f(x) = ax² + bx. The five intervals below all have the same length h and sit side by side, starting at x = 0. The graph on the left shows the chords. The graph on the right plots each interval's average rate of change at the interval's midpoint.
| interval | average rate of change | change from the previous |
|---|---|---|
| [0.0, 1.0] | 1.00 | — |
| [1.0, 2.0] | 3.00 | +2.00 |
| [2.0, 3.0] | 5.00 | +2.00 |
| [3.0, 4.0] | 7.00 | +2.00 |
| [4.0, 5.0] | 9.00 | +2.00 |
The average rate of change of a function f over the closed interval [a, b] is the slope of the secant line from the point (a, f(a)) to the point (b, f(b)). Rise over run on that line is the change in output over the change in input.
Drag the two endpoints. The dashed line through the two points on the curve is the secant line. Its slope, rise over run, is exactly the average rate of change.
Now ask a second question: if you list the average rates over consecutive equal-length intervals, how are those rates changing?
For a linear function the rates are all the same constant, so they are changing at a rate of zero. For a quadratic function the rates fit a linear function, so they are changing at a constant rate. In fact, when the intervals have length h, each rate is 2ah larger than the one before it.
Pick a function and an interval length h. The table finds the average rate of change on each consecutive interval of length h, and then asks the new question: how are those rates changing?
| x | 0 | 1 | 2 | 3 | 4 | 5 |
| f(x) | 0.00 | 1.00 | 4.00 | 9.00 | 16.00 | 25.00 |
| average rate on the interval ending here | — | 1.00 | 3.00 | 5.00 | 7.00 | 9.00 |
| change in the rate | — | — | +2.00 | +2.00 | +2.00 | +2.00 |
When the average rate of change over equal-length intervals is increasing for all small-length intervals, the graph is concave up. When it is decreasing for all small-length intervals, the graph is concave down. Concavity reads the trend of the rates, so it works for any function, not only quadratics.
Each chord below is colored by what the average rates are doing: green where the rates are increasing from one interval to the next, coral where they are decreasing. All intervals have the same length.
| interval | average rate | compared with the previous |
|---|---|---|
| [0.0, 1.0] | 1.00 | — |
| [1.0, 2.0] | 3.00 | ▲ increasing |
| [2.0, 3.0] | 5.00 | ▲ increasing |
| [3.0, 4.0] | 7.00 | ▲ increasing |
| [4.0, 5.0] | 9.00 | ▲ increasing |
| [5.0, 6.0] | 11.00 | ▲ increasing |
Given a table with equally spaced inputs, you can tell what kind of function it could come from. Find the average rate on each interval. If the rates are all equal, the data is linear. If the rates are not equal but they change by the same amount each step, the data is quadratic. If neither happens, it is something else.
| Function type | Rates over consecutive equal-length intervals | How the rates change |
|---|---|---|
| Linear | The same constant | At a rate of 0 |
| Quadratic | Given by a linear function | At a constant rate, 2ah (concave up if a > 0, down if a < 0) |
| Other types | Neither of these | Not at a constant rate; may be concave up, down, or both |
Each table has equally spaced inputs. Work out the average rates and the changes in the rates, then decide what kind of function could be behind the data.
| seconds | 0 | 1 | 2 | 3 | 4 |
| height (ft) | 0 | 48 | 64 | 48 | 0 |
| average rate | — | 48.00 | 16.00 | -16.00 | -48.00 |
| change in the rate | — | — | -32.00 | -32.00 | -32.00 |