Set the function equal to zero, factor the polynomial (if possible), and solve for $x$. Use the quadratic formula if it's a quadratic.

Factor the numerator and denominator. Simplify the rational function. Set the denominator equal to zero and solve for $x$.

Compare the degrees of the numerator and denominator. If numerator degree < denominator degree, $y=0$. If degrees are equal, $y=$ ratio of leading coefficients. If numerator degree > denominator degree, there is no horizontal asymptote.

Factor the numerator and denominator. Identify common factors that cancel out. The $x$-value that makes the canceled factor zero is the location of the hole.

Identify the degree and leading coefficient. If the degree is even and the leading coefficient is positive, both ends go up. If the degree is even and the leading coefficient is negative, both ends go down. If the degree is odd and the leading coefficient is positive, the left end goes down, and the right end goes up. If the degree is odd and the leading coefficient is negative, the left end goes up, and the right end goes down.

Find zeros, vertical asymptotes, horizontal asymptote, and holes. Plot these features on a coordinate plane. Determine the sign of the function in each interval created by the zeros and vertical asymptotes. Sketch the graph based on this information.

Calculate the function's value at the endpoints of the interval, $f(a)$ and $f(b)$. Use the formula: $\frac{f(b) - f(a)}{b - a}$.

If the polynomial has real coefficients and the discriminant ($b^2 - 4ac$) of a quadratic factor is negative, then the function has complex zeros.

If $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$. Use synthetic division to divide the polynomial by $(x-a)$ and find the remaining factors.

If you know one factor of the polynomial, divide the polynomial by that factor using long division. The quotient will be a polynomial of lower degree, which may be easier to factor to find the remaining zeros.

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

$f(x) = \frac{p(x)}{q(x)}$, 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.

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.