Topic 1.3 Rates of Change in Linear and Quadratic Functions: characterize linear functions by their constant rate of change and quadratic functions by their constant rate of change of the rate of change.

What this topic is asking

The College Board (Topic 1.3) wants you to characterize two function families by their rate-of-change behavior. A linear function has a constant rate of change: its first differences over equally spaced inputs are constant. A quadratic function has a rate of change that is itself changing at a constant rate: its second differences are constant. Being able to classify a table or context this way is a recurring exam skill.

Linear functions: constant first differences

This is the defining feature: a quantity that changes by the same amount for each equal step in the input is linear. Simple interest, a fixed monthly fee, and constant-speed motion are all linear.

Quadratic functions: constant second differences

So linear functions are flagged by constant first differences and quadratics by constant second differences. This nested pattern, where each "level" of differences strips away one degree, continues for higher-degree polynomials in Topic 1.4.

Classifying data from a table

The exam often gives a table and asks for the function type. The routine is mechanical when the inputs are equally spaced:

1. Compute first differences. Constant means linear; stop.
2. If not constant, compute second differences. Constant means quadratic.
3. If neither is constant, check ratios for exponential behavior (Topic 2.2).

Why the difference test works

The difference test works because of how polynomial outputs accumulate. For a line $mx + b$, stepping the input by $\Delta x$ always adds $m \Delta x$ to the output, so first differences are constant. For a quadratic $ax^2 + bx + c$, the amount added per step grows linearly with $x$, so first differences form a linear sequence, and the differences of a linear sequence are constant, giving constant second differences. Each extra degree of the polynomial pushes the constancy down one more level of differencing, which is the principle behind Topic 1.4's claim that a degree-$n$ polynomial has constant $n$th differences. Recognizing this structure means you can both classify a table and predict how many levels of differencing you would need for any polynomial degree, a connection the exam likes to probe.

Try this

Q1. A table has constant first differences of $-4$. What type of function is it, and is it increasing or decreasing? [1 point]

• Cue. Linear (constant first differences); decreasing, since the slope $-4$ is negative.

Q2. A table has first differences $1, 4, 7, 10$. What type of function is it? [1 point]

• Cue. Second differences are $3, 3, 3$ (constant), so it is quadratic.