Rates of change in linear and quadratic functions

A linear function changes at a constant rate: over equal-length input intervals, the output changes by equal amounts. That is the whole identity of a line. In $f(x) = mx + b$, the rate is $m$, and it is the same whether you measure over $[0, 1]$, $[50, 54]$, or any interval at all.

In a table with equally spaced inputs, this appears as constant first differences: subtract each output from the next, and every answer matches. The intercept $b$ never touches the differences; it shifts where the line sits, not how fast it climbs.

A quadratic function does not have one rate, but its rates follow the simplest possible pattern: over consecutive equal-length intervals, the average rates of change themselves change by a constant amount. For $f(x) = x^2$, the average rates over $[1,3]$, $[3,5]$, $[5,7]$ are $4$, $8$, $12$: different, but stepping up by exactly $4$ each time.

In a table this appears one layer down: the first differences are not constant, but the second differences, the differences of the differences, are. Constant second differences over equally spaced inputs is the quadratic signature, exactly as constant first differences is the linear one.

Given a table with equally spaced inputs, run this every time:

1. Write the first differences between consecutive outputs. All equal? Linear, and that shared value per step is the rate.
2. Not equal? Write the second differences. All equal (and not zero)? Quadratic.
3. Read direction and bend from the layers: positive first differences mean increasing; first differences growing means concave up, shrinking means concave down.

Two cautions. The inputs must be equally spaced or the pattern lies. And a pattern in the differences does not mean the function is linear; it is the differences being constant, not merely patterned, that certifies a line.

A parabola opening upward is concave up because its rate of change is always increasing: climbing rates on the way up, or falls that shrink toward the vertex and turn to gains. Opening downward is the mirror story, a rate that decreases steadily through the vertex, from gains to losses.

This is why a thrown ball’s height, with average rates over successive seconds of $15, 5, -5$ meters per second, is quadratic behavior: the rates drop by the same $10$ each second. The function’s values rise and then fall; the rate marches steadily downward the whole time. Keeping those two layers separate is most of the battle in this topic.