Mathfolis

Polynomial Functions and Rates of Change

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

Topic 1.4 extends the difference idea from quadratics to all polynomials. The n-th finite differences of a degree-n polynomial are constant when inputs are equally spaced — the discrete analog of the calculus statement that the n-th derivative of a degree-n polynomial is constant. A polynomial of degree n has at most n − 1 turning points, and its AROC over [p, p + h] is a polynomial in p of degree n − 1.

Degree from differences (equally spaced inputs)

Degree 1 (linear)
1st differences constant\text{1st differences constant}
Degree 2 (quadratic)
2nd differences constant\text{2nd differences constant}
Degree n
n-th differences constant\text{n-th differences constant}

Turning points

Bound
degree n    ⟹    at most n−1 turning points\text{degree } n \;\implies\; \text{at most } n - 1 \text{ turning points}
Caution: 'At most n − 1' turning points — not exactly. f(x)=x3f(x) = x^3 is degree 3 yet has zero turning points (it is strictly increasing).
Type 1

Identify degree from constant nth differences

Compute successive difference columns until you find one that is constant. The column number equals the degree.

Example 1
A function has values f(0)=0f(0) = 0, f(1)=1f(1) = 1, f(2)=8f(2) = 8, f(3)=27f(3) = 27, f(4)=64f(4) = 64, f(5)=125f(5) = 125 at equally spaced inputs. What is its degree? (A) 1 (B) 2 (C) 3 (D) 4

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Type 2

Relate degree to turning points

Use the bound 'at most n − 1 turning points for degree n' in both directions: given degree, bound the turning-point count; given turning points, bound the degree.

Example 2
A polynomial graph has 2 visible turning points. What is the smallest possible degree? (A) 2 (B) 3 (C) 4 (D) 5

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Type 3

AROC pattern of a polynomial

Compute AROC of a polynomial over several equally spaced unit intervals. The AROCs form a sequence; analyze its difference column to identify the degree of the original polynomial.

Example 3
For f(x)=x3f(x) = x^3, the AROCs over [0,1][0, 1], [1,2][1, 2], [2,3][2, 3] are 1,7,191, 7, 19. What is the degree of the AROC pattern in the left endpoint p? (A) 0 (constant) (B) 1 (linear) (C) 2 (quadratic) (D) 3 (cubic)

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