Precalculus · Unit 1: Polynomial and Rational Functions · 14 min read · Updated 2026-05-11
Polynomial functions and rates of change — AP Precalculus
AP Precalculus · Unit 1: Polynomial and Rational Functions · 14 min read
1. Core Concepts of Polynomial Rates of Change★★☆☆☆⏱ 3 min
This topic explores how the output of a polynomial changes as input changes, connecting algebraic polynomial properties to calculus-based behavior core to the AP Precalculus curriculum. It contributes 4–6% of total AP exam score, appearing in both multiple-choice and free-response sections.
2. Average Rate of Change of Polynomials★★☆☆☆⏱ 4 min
The average rate of change (AROC) of a function over $[x_1, x_2]$ describes the overall rate the function's output changes across the entire interval. Geometrically, it equals the slope of the secant line connecting the two endpoints on the function graph.
A key property: for linear (degree 1) polynomials, AROC is constant over any interval. For all higher degree polynomials, AROC changes with the interval chosen, matching the curved shape of the graph. AROC can also be calculated from tables even if the full function is unknown.
3. Instantaneous Rate of Change and Difference Quotients★★★☆☆⏱ 4 min
The instantaneous rate of change (IROC) at $x=a$ describes the rate of change at that exact input, rather than across an interval. It is defined as the limit of the average rate of change as the interval width approaches zero, and equals the slope of the tangent line at $x=a$.
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
The expression $\frac{f(a + h) - f(a)}{h}$ is called the difference quotient, the core expression for derivatives from first principles. All polynomials are differentiable everywhere, so IROC exists for every real input. The AP exam may require you to use the difference quotient method on free-response questions, so both this method and direct differentiation with the power rule must be mastered.
4. Analyzing Polynomial Behavior with Rates of Change★★★☆☆⏱ 3 min
For a degree $n$ polynomial $f(x)$, the first derivative $f'(x)$ is degree $n-1$, and the second derivative $f''(x)$ (derivative of $f'$) is degree $n-2$. We use the sign of these derivatives to identify key features of the original polynomial:
$f'(x) > 0$ on interval: $f(x)$ is increasing
$f'(x) < 0$ on interval: $f(x)$ is decreasing
Local extremum: $f'(x) = 0$ *and* $f'(x)$ changes sign at the point
$f''(x) > 0$ on interval: $f(x)$ is concave up (rate of change is increasing)
$f''(x) < 0$ on interval: $f(x)$ is concave down (rate of change is decreasing)
Inflection point: $f''(x)$ changes sign at the point (concavity changes)
Common Pitfalls
Why: Rushing to answer leads to mismatched endpoint order, resulting in a sign error
Why: Trying to cancel immediately leads to incorrect cancellation of constant terms
Why: Assuming all critical points are extrema, but $f(x)=x^3$ has $f'(0)=0$ with no sign change and no extremum
Why: The phrase 'average rate of change over the entire interval' sounds like it requires averaging smaller intervals, which is wrong
Why: Confusing the degree of the derivative: derivative of cubic is quadratic, which is not constant