Quotient of two polynomial functions p(x)/q(x) (where q is not a factor of p),
or call them N(x)/D(x) for numerator and denominator polynomials.

Glance below for some examples.

The denominator q, or D, can not be a factor of p, or N, because otherwise this would be a polynomial.

Glance below for some examples.

The denominator q, or D, can not be a factor of p, or N, because otherwise this would be a polynomial.

**Domain** of rational function is R minus the values that make q(x) denominator zero.

**x-intercepts** are the zeros of the numerator p(x), i.e. p(x)=0 Solve.

**y-intercept**: set x to 0, if possible.

**Vertical asymptotes** at x's that make the simplified (i.e. after all cancellations
between numerator and denominator) denominator 0.
n-degree denominator means ≤n vertical asymptotes: [0..n].
Note: non-rational functions can have infinite vertical asymptotes, e.g. *tan*

**Horizontal asymptote**: 0 or 1.

Ratio of dominant/leading terms of numerator and denominator:

Degrees of N and D are the same:

ax^{n}/bx^{n} --> y=a/b

x^{n}/x^{n} --> y=1 Special case when a=b=1.

Degree of N is greater than degree of D:

x^{i}/x^{j} i**>**j --> none.

Except see below for oblique asymptote if degree of N is one more than degree of D.

Degree of N is less than degree of D:

x^{i}/x^{j} i**<**j --> y=0

Graph can cross horizontal or oblique asymptote but not vertical asymptotes.

Horizontal asymptote is the limit as x approaches ±∞.

Note: non-rational functions can have 2 horizontal asymptotes:

*arctan* has
at ±π/2.

x/√(x^{2}+1) has at ±1

CDF of standard normal function has at 0 and 1

Examples:

Simplest: y=1/x

Open-top box with square base x, height y of given volume V.

Surface area (of outside or inside, not both) is S(x)=(x^{3}+4V)/x

Cylinder of radius r and volume V.

Surface area: SA(r)=(2πr^{3}+2V)/r

**Oblique/slant asymptote**: 0 or 1.

If degree of numerator is one more than degree of denominator:

x^{i+1}/x^{i}

Only one of horizontal and oblique asymptotes are possible.

Divide N by D to get a linear quotient which is the oblique line, ignore remainder.