End behavior describes what happens to the function's values as $x$ approaches positive or negative infinity. It's determined by the leading term (degree and leading coefficient).

The degree determines the maximum number of turning points and the end behavior. Even degree: ends go in the same direction. Odd degree: ends go in opposite directions.

The sign of the leading coefficient determines whether the graph rises or falls as $x$ approaches positive or negative infinity. Positive: rises to the right. Negative: falls to the right.

If $x = a$ is a zero of a polynomial, then $(x - a)$ is a factor of the polynomial.

Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not. They indicate values of $x$ where the function approaches infinity.

Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and denominator.

Holes occur when a factor cancels out in both the numerator and denominator. The function is undefined at that $x$-value, but the limit exists.

Compare the degrees of the numerator and denominator. If the degree of the denominator is greater, $y=0$. If the degrees are equal, $y$ is the ratio of leading coefficients.

The rate of change (slope) is constant for linear functions, meaning the function increases or decreases at a steady rate.

The rate of change varies for quadratic functions, meaning the function's increase or decrease is not constant.

A sum of terms, each consisting of a coefficient and a variable raised to a non-negative integer power.

The highest power of the variable in the polynomial.

A ratio of two polynomial functions, expressed as $f(x) = p(x)/q(x)$.

A vertical line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$.

A horizontal line $y = b$ that the function approaches as $x$ approaches infinity or negative infinity.

A point where the function is undefined because a factor cancels out in both the numerator and denominator.

How quickly a function's output changes with respect to its input.

Zeros of a polynomial function that are complex numbers.

The coefficient of the term with the highest power in a polynomial.

The behavior of a function as $x$ approaches positive or negative infinity.

Polynomials: No breaks or asymptotes. | Rational Functions: Can have vertical/horizontal asymptotes and holes.

Vertical Asymptotes: Function approaches infinity. | Holes: Function is undefined, but limit exists.

Even Degree: Ends go in the same direction. | Odd Degree: Ends go in opposite directions.

Linear: Constant rate of change (slope). | Quadratic: Rate of change varies; not constant.

Multiplicity 1: The graph crosses the x-axis at the zero. | Multiplicity 2: The graph touches the x-axis and turns around at the zero.

Numerator < Denominator: y=0. | Numerator = Denominator: y = ratio of leading coefficients. | Numerator > Denominator: No horizontal asymptote.

Positive Leading Coefficient: Graph rises to the right. | Negative Leading Coefficient: Graph falls to the right.

Real Zeros: Intersect the x-axis. | Complex Zeros: Do not intersect the x-axis.

Factoring: Useful for simple polynomials. | Quadratic Formula: Used when factoring is not possible (specifically for quadratics).

Without Holes: Continuous except at asymptotes. | With Holes: Discontinuity at the location of the hole.