AP Precalculus Unit 1 Notes: Understanding Polynomial Functions

Rate of change

A measure of how much the output of a function changes when the input changes.

Average rate of change

The rate of change of a function over an interval [a,b], computed as (f(b)-f(a))/(b-a).

Secant line

A line that passes through two points on a graph; its slope equals the average rate of change over that interval.

Linear function

A function of the form $f(x)=mx+b$ that has a constant rate of change and a straight-line graph.

Slope (m)

The constant rate of change of a linear function; the change in y per 1 unit change in x.

y-intercept (b)

The value of a linear function when x=0; the point where the graph crosses the y-axis.

Nonlinear function

A function whose average rate of change depends on the interval chosen (its rate of change is not constant).

Increasing (function behavior)

A function whose outputs go up as inputs go up; this does not necessarily mean the rate of change is constant.

Polynomial function

A function that can be written as a sum of terms $a_k x^k$ with nonnegative integer exponents $k$ and real coefficients.

Coefficient

A real-number multiplier of a power of $x$ in a polynomial (e.g., $a_n$ in $a_n x^n$).

Degree (of a polynomial)

The highest exponent of x with a nonzero coefficient in a polynomial.

Leading term

The highest-degree term of a polynomial, $a_n x^n$.

Leading coefficient

The coefficient of the leading term ($a_n$); it strongly influences end behavior.

Turning point

A point on a polynomial’s graph where it changes from increasing to decreasing or vice versa; a degree $n$ polynomial can have at most $n-1$ turning points.

End behavior

The way the left and right “tails” of a polynomial’s graph behave as $x→∞$ or $x→-∞$, determined by the leading term.

Parity of degree

Whether the polynomial’s degree is even or odd, which helps predict whether both ends go the same direction (even) or opposite directions (odd).

Zero (root)

An x-value r such that f(r)=0; graphically, it corresponds to an x-intercept.

x-intercept

A point where the graph meets the x-axis (where $y=0$), corresponding to a zero of the function.

Factor (as it relates to zeros)

If ($x-r$) is a factor of $f(x)$, then $r$ is a zero of $f(x)$ because $f(r)=0$.

Multiplicity

The number of times a zero is repeated; if $x=r$ has multiplicity $k$, then ($x-r$)$^k$ is a factor of the polynomial.

Odd multiplicity

A zero multiplicity that makes the graph cross the x-axis at that intercept (the sign of the function changes).

Even multiplicity

A zero multiplicity that makes the graph touch the x-axis and turn around (the sign of the function does not change).

Fundamental Theorem of Algebra (counting idea)

A polynomial of degree $n$ has exactly $n$ complex zeros when counted with multiplicity.

Polynomial inequality

An inequality involving a polynomial (e.g., $f(x)>0$ or $f(x)≤0$) asking where the polynomial is positive, negative, or zero.

Sign chart (sign analysis)

A method for solving polynomial inequalities by factoring, finding zeros (critical points), testing intervals, and determining where the polynomial is positive/negative and whether to include zeros.