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:
axn/bxn --> y=a/b
xn/xn --> y=1 Special case when a=b=1.
Degree of N is greater than degree of D:
xi/xj 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:
xi/xj 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/√(x2+1) has at ±1
CDF of standard normal function has at 0 and 1
Open-top box with square base x, height y of given volume V.
Surface area (of outside or inside, not both) is S(x)=(x3+4V)/x
Cylinder of radius r and volume V.
Surface area: SA(r)=(2πr3+2V)/r
Oblique/slant asymptote: 0 or 1.
If degree of numerator is one more than degree of denominator:
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.