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.

$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$

$f(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials.

$\frac{f(b) - f(a)}{b - a}$

$m = \frac{y_2 - y_1}{x_2 - x_1}$

If $f(x) = \frac{ax^n + ...}{bx^n + ...}$, then $y = \frac{a}{b}$

$y = 0$

There is no horizontal asymptote. Consider long division to find slant asymptote if the degree of the numerator is exactly one more than the degree of the denominator.

Set $f(x) = 0$ and solve for $x$.

Set the denominator $q(x) = 0$ and solve for $x$, excluding any values that are also zeros of the numerator.

Find common factors in the numerator and denominator. The $x$-value where the factor equals zero is the location of the hole.

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.