AP PRECALCULUS — UNIT 1A · POLYNOMIAL & RATIONAL FUNCTIONS
1.3 — Rates of Change in Linear and Quadratic Functions
Notes — Differences That Reveal a Function's Type
💡 Learning Objectives (1.3.A and 1.3.B) By the end of this lesson you will be able to:
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1. Why Linear Functions Are Special
A linear function has a constant rate of change. That is exactly what its slope captures. Pick any two points on the line of y = mx + b and the AROC works out to m, every single time.
💡 Linear functions: constant AROC If f(x) = mx + b, then for any a < b in its domain, AROC of f on [a, b] = ( (mb + b) − (ma + b) ) / (b − a) = m. The output answer doesn't depend on which interval you choose. |
For a linear function over equally spaced inputs, the differences between consecutive outputs (the ‘first differences’) are constant. Here every step adds 2.
📘 Example — Constant first differences For f(x) = 3x − 1 sampled at x = 0, 1, 2, 3, 4: outputs are −1, 2, 5, 8, 11. Each consecutive difference is 3 — the slope. |
2. Quadratic Functions: AROC Itself Is Linear
For a quadratic function, the average rate of change varies from interval to interval — but it varies in a very predictable way.
💡 AROC of a quadratic over equal-length intervals If f(x) = ax² + bx + c and we compute AROC on each unit interval [n, n+1] for n = 0, 1, 2, …, the resulting sequence of AROC values is itself a linear function of n. Equivalently, the second differences of the original outputs are constant. |
For g(x) = x² sampled at x = 0, 1, 2, 3, 4, 5: the first differences are 1, 3, 5, 7, 9 — themselves a linear sequence with common difference 2. So the second differences are constant.
📘 Example — Walking through it For g(x) = x² + 2x: Outputs at x = 0,1,2,3,4: 0, 3, 8, 15, 24. 1st differences: 3, 5, 7, 9. ← linear (rises by 2 each step) 2nd differences: 2, 2, 2. ← constant ⇒ confirms quadratic, with leading coefficient ½ × 2 = 1. |
2.1 Using differences to recover the leading coefficient
If you sample f(x) = ax² + bx + c at integer x-values, the constant value of the second differences is 2a. Knowing this lets you read the leading coefficient straight off the difference table.
3. Generalising — How to Detect a Polynomial's Degree
💡 Degree from successive differences Sample f(x) at equally spaced inputs. Take repeated differences:
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4. The Connection to Concavity
The pattern of AROC values over equal intervals tells you about concavity:
- If the AROC values are increasing as x grows, the graph is bending upward — concave up.
- If the AROC values are decreasing as x grows, the graph is bending downward — concave down.
- If the AROC values are constant (a linear function), the graph has no curvature — it is a straight line and so neither concave up nor concave down.
Concave up curves have AROC values that grow as you move from one interval to the next; concave down curves have AROC values that shrink.
📘 Example — Reading concavity from a difference table Sample data: x: 0, 1, 2, 3, 4 y: 10, 9, 6, 1, −6 1st differences: −1, −3, −5, −7 — all negative, so y is decreasing on every step. 2nd differences: −2, −2, −2 — constant negative. Constant negative second differences mean the AROCs are decreasing → graph is concave down. The function is quadratic with leading coefficient ½ × (−2) = −1. |
5. A Quick Diagnostic Routine
- Step 1: List outputs at equally spaced inputs.
- Step 2: Compute first differences. Constant? Linear, slope = that constant.
- Step 3: If not constant, compute second differences. Constant? Quadratic, leading coefficient = (constant)/2.
- Step 4: If still not constant, the data could come from a higher-degree polynomial — keep going.
⚠️ Inputs must be equally spaced The constant-difference test only works if the inputs in your table are evenly spaced. If they aren't, compute AROCs over equal-width intervals first, or interpolate. |
6. Summary
- Linear functions have constant AROC over any interval — the slope
- Quadratic functions have AROC values that form a linear pattern themselves
- So linear ⇔ first differences constant; quadratic ⇔ second differences constant
- AROC values that are increasing signal concave up; decreasing signal concave down
- These tools let you classify the type of relationship from a table of values alone