Functions to know
 linears, quadratics, polynomials, rationals
 exponetials, logs
 sine, cosine, tangent, secant, cosecant, cotangent, and corresponding inverses
General
Function properties
 average rate of change (i.e. average slope) of a continuous function between two given end points: depends only on the values of the function at the end points!
 even/odd
Property  Examples  

Odd  {$f(x) = f(x)$}  {$x$}, {$x^3$}, {$x^{\frac{1}{3}}$}, {$\sin x$}, {$\tan x$}, {$\arcsin x$}, {$x\cos x$}, {$\frac{x^2+1}{x}$} 
Even  {$f(x) = f(x)$}  {$1$}, {$x$}, {$x^2$}, {$x^4$}, {$\cos x$}, {$x\sin x$}, {$e^{x^2}$}, {$\frac{\sin x}{x}$} 
Neither  {$x+x^2$}, {$\ln x$}, {$e^x$}, {$\sqrt{x}$} 
Graphs
 shifting, scaling, reflecting: {$f(x)$} → {$c f(\frac{xa}{b}) + d$}
 scaled vertically by {$c$} (negative => vertical reflection)
 scaled horizontally by {$b$} (negative => horizontal reflection)
 shifted up by {$d$} (negative => shifted down)
 shifted right by {$a$} (negative => shifted left)
 properties: end behavior, intercepts, zeros, asymptotes, holes, amplitudes, periods, frequency, increasing/decreasing intervals (doesn't matter if you include endpoints)
Domain problems
 {$\sqrt{4x^2}$}, {$\sqrt{x^24}$} – can do directly or use sign analysis
 {$\frac{x+4}{x^24}$}, {$\frac{1}{\sqrt{x^24}}$}, {$\log(1x^2)$}, {$e^\frac{1}{1x}$}
 {$\sqrt{\frac{x+4}{x^24}}$} – requires sign analysis of rational function!
 {$\ln\left(\frac{x+4}{x^24}\right)$} – requires sign analysis of rational function!
 Compare to the domain of {$\ln(x+4)  \ln(x^24)$}
 Domain of {$f \circ g$} where {$f(x) = \frac{1}{x}$} and {$g(x) = \frac{1}{x1}$}. Note that {$(f \circ g)(x) = x1$}, but the domain of {$f \circ g$} does not include {$x=1$} since it is not in the domain of {$g$}.
Onetoone and inverses
 {$(f\circ f^{1})(x) = f(f^{1}(x)) = x$}
 {$(f^{1}\circ f)(x) = f^{1}(f(x)) = x$}
 {$f$} and {$f^{1}$} swap domains and ranges (i.e. can compute the range of {$f$} by computing the domain of {$f^{1}$}).
 The graph is reflected about {$y=x$}.
 Inverse of {$\sqrt[3]{x^51}$}, {$\sqrt[3]{\arcsin(x)1}$}, {$\sqrt[3]{e^x1}$}, {$\sqrt[3]{\log_5 x1}$}
 Inverse of {$\frac{x+1}{x1}$}, {$\log\left(\frac{x+1}{x1}\right)$}, {$\arctan\left(\frac{x+1}{x1}\right)$}
Polynomials and Rationals
Linear functions
 {$f(x) = mx + b$}
 slope ({$m$}), intercept ({$b$})
 parallel ({$m x + k$}) and perpendicular ({$\frac{1}{m} x + k$}) lines
Quadratics
 {$f(x) = ax^2 + bx + c$} ↔ {$f(x) = a(xh)^2 + k$} (standard form)
 {$a>0$} ⇒ opens upward and has a min, {$a<0$} ⇒ opens downward and has a max
 vertex {$(h,k)$} (min or max, depending on the sign of {$a$})
 completing the square
 pattern: {$x^2 + (2d)x = x^2 + (2d)x + d^2  d^2 = (x+d)^2d^2$}
 pattern: {$a x^2 + b x = a(x^2 + \frac{b}{a} x)$} and work on what is inside the parentheses as above
Polynomials
Polynomial of degree {$n$}: {$f(x)=a_n x^n + a_{n1} x^{n1} + … + a_{1}x + a_{0}$}, {$a_n \ne 0$}
 End behavior determined by leading term
term  As {$x\rightarrow \infty$}  As {$x\rightarrow \infty$} 

{$f(x)=x^n+…\qquad$}, {$n$} even  {$\infty$}  {$\infty$} 
{$f(x)=x^n+…\qquad$}, {$n$} odd  {$\infty$}  {$\infty$} 
{$f(x)=x^n+…\qquad$}, {$n$} even  {$\infty$}  {$\infty$} 
{$f(x)=x^n+…\qquad$}, {$n$} odd  {$\infty$}  {$\infty$} 
Note how all the infinities invert when the coefficient was negative.
Zeros of polynomials
 Intermediate value theorem for a polynomial {$f$}: If {$f(a)$} and {$f(b)$} have opposite signs, then {$f(c)=0$} for some {$c$} between {$a$} and {$b$}.
 Equivalent statements for {$f$} polynomial
 {$c$} is a zero of {$f$}
 {$c$} is an {$x$}intercept of {$f$}
 {$x=c$} is a solution to {$f(x)=0$}; i.e. {$f(c)=0$}
 {$xc$} is a factor of {$f(x)$} (Important!)
 Multiplicity of the zero determines how the polynomial hits the {$x$}axis – very useful for drawing graphs quickly without having to do any sign analysis!
 Rational Zeros Theorem: {$P(x)=a_n x^n + a_{n1} x^{n1} + … + a_{1}x + a_{0}$}, {$a_n \ne 0$}
 Every rational zero of {$P(x)$} is of the form {$\pm\frac{p}{q}$} where {$p$} is a factor of {$a_0$} and {$q$} is a factor of {$a_n$}.
Long division and synthetic division
 Useful for figuring out zeros
 Divide {$P(x)$} by {$D(x)\ne 0$}, then
 unique decomposition: {$P(x) = D(x) \cdot Q(x) + R(x)$} where {$R(x)$} is zero or has degree less than {$D(x)$}
 If {$D(x) = xc$}, then the remainder {$R(x) = P(c)$} (i.e. the remainder is {$P$} evaluated at {$c$}).
 unique decomposition: {$\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}$}
 {$Q(x)$} gives slant/curved asymptote of the rational function {$\frac{P(x)}{D(x)}$}
 unique decomposition: {$P(x) = D(x) \cdot Q(x) + R(x)$} where {$R(x)$} is zero or has degree less than {$D(x)$}
 Synthetic division can be used to quickly divide {$P$} by {$(xc)$} to simultaneously test if {$c$} is a factor and to obtain a factorization.
Rational functions
To analyze, {$f(x) = \frac{(x1)(2x+3)(x2)}{(x3)^2(x1)}$}
 Cancel out holes first: {$f(x) = \frac{(2x+3)(x2)}{(x3)^2}\cdot\frac{x1}{x1} = \frac{(2x+3)(x2)}{(x3)^2}$}
 holes: {$x = 1$}
 zeros: {$x = \frac{3}{2}, 2$}
 vertical asymptotes: {$x = 3$}
 Same degree in the numerator and denominator ⇒ horizontal asymptote: {$y = 2$}
 If the degree of the denominator is larger, then the asymptote is 0.
 If the degree of the numerator is larger, then there is a slant or curved asymptote to be determined by long division.
 Again, the multiplicity of both the zeros and the vertical asymptotes can help graph the function very quickly without function/asymptote sign analysis.
Exponentials and Logs
All the properties on the right column are gotten by taking the equations on the left column and taking the {$\log_a$} of both sides, reordering, and using {$X=a^x$} and {$Y=a^y$}.
Properties  Exponential  Log 

Anything to the power of 0 is 1  {$a^0 = 1$}  {$\log_a 1 = 0$} 
Anything to the power of 1 is itself  {$a^1 = a$}  {$\log_a a = 1$} 
Multiply powers ⇒ add exponents  {$a^x a^y = a^{x+y}$}  {$\log_a (X Y) = \log_a X+\log_a Y$} 
Divide powers ⇒ subtract exponents  {$\frac{a^x}{a^y} = a^{xy}$}  {$\log_a (\frac{X}{Y}) = \log_a X  \log_a Y$} 
Inverses  {$a^{\log_a X} = X$}  {$\log_a a^x = x$} 
Bring down the exponent  {$a^{y \log_a X} = (a^{\log_a X})^y = X^y$}  {$y \log_a X = \log_a (X^y)$} 
Change of base  {$a^x = (b^{\log_b a})^x$}  {$\log_b X = \frac{\log_a X}{\log_a b}$} 
You are not required to know the highlighted ones but they help you understand the bigger picture.
Applications
 Exponential growth
 simple interest, compound interest: {$A(t)=P \big[\underbrace{\left(1+\frac{r}{n}\right)^{n}}_\text{1+APY}\big]^{t}$}
 continuously compounded interest: {$A(t)=Pe^{rt}$}
 population growth (including continuous growth, which has the same form as continuously compounded interest)
 Exponential decay
 half life: {$m(t) = m_0 \left(\frac{1}{2}\right)^{t/t_h}$}, where {$t_h$} is the half life
 Newton's law of cooling (formula would be provided if applicable): {$T(t) = T_s + (T_0T_s)e^{kt}$}.
 Logarithms (formulas would be provided if applicable)
 {$pH = \log[H^+]$}
 Richter scale: {$M=\log\frac{I}{S}$}
 Decibels
Trigonometric functions

Unit circle or right triangle
{$\cos t = x = \frac{adj}{hyp}$} {$\sec t = 1/\cos t$} {$\sin t = y = \frac{opp}{hyp}$} {$\csc t = 1/\sin t$} {$\tan t = \frac{y}{x} = \frac{opp}{adj}$} (slope) {$\cot t = 1/\tan t$}  Know the quadrant labels.
 Know and recognize the sines, cosines, and tangents of {$0, \pi/6, \pi/4, \pi/3, \pi/2$}, and the corresponding angles in the other quadrants.
 {$a\sin(k(xb))$}, {$a\cos(k(xb))$}, {$a\tan(k(xb))$}
 amplitude {$a$} (cos or sin)
 phase shift {$b$}
 period {$2\pi/k$} (cos or sin) OR {$\pi/k$} (tan)
 frequency = 1/period = {$k/2\pi$} (cos or sin) OR {$k/\pi$} (tan)
 vertical asymptotes {$\pi/2 + \pi k$} (tan)
Inverses
function  domain  range 

{$\arccos x = \cos^{1} x$}  {$ [1, 1] $}  {$ [0, \pi] $} 
{$\arcsin x = \sin^{1} x$}  {$ [1, 1] $}  {$ [\pi/2, \pi/2] $} 
{$\arctan x = \tan^{1} x$}  {$(\infty, \infty)$}  {$(\pi/2, \pi/2)$} 
 Be able to compute the following exactly
 {$\arcsin(\sin(\frac{200\pi}{3}))$}, {$\arccos(\cos(\frac{201\pi}{6}))$} – figure out terminal point on the unit circle
 {$\cos(\arctan(20))$}, {$\sin(\arccos(\frac{1}{4}))$}, {$\tan(\arcsin(x))$} – draw a triangle and figure out the lengths
Trigonometric algebra

Be able to simplify expressions, prove identities, and solve equations.

Identities (know the first column of formulas)
Important ones  Related ones  

Pythagorean  {$\cos^2 t + \sin^2 t = 1$}  {$1 + \tan^2 t = \sec^2 t$} {$\cot^2 t + 1 = \csc^2 t$} 
Even/odd  {$\cos(t) = \cos t$} {$\sin(t) = \sin t$} {$\tan(t) = \tan t$} 
similarly for {$\sec, \csc, \cot$} 
Cofunction  {$\cos\left(\frac{\pi}{2}t\right) = \sin t$} {$\sin\left(\frac{\pi}{2}t\right) = \cos t$} 
similarly for {$\sec/\csc, \tan/\cot$} 
Angle sum  {$\cos(s+t) = \cos s \cos t  \sin s \sin t$} {$\sin(s+t) = \sin s \cos t + \cos s \sin t$} 
{$\cos(st) = \cos s \cos t + \sin s \sin t$} {$\sin(st) = \sin s \cos t  \cos s \sin t$} get {$\tan$} from {$\cos$} and {$\sin$} 
Double angle  {$\cos(2t) = \cos^2 t  \sin^2 t$} {$\qquad= 2\cos^2 t  1$} {$\qquad= 1  2\sin^2 t$} {$\sin(2t) = 2 \sin t \cos t$} 
get {$\tan$} from {$\cos$} and {$\sin$} 
 Double angle formula useful for lowering degree
(Don't memorize these! Always figure them out from formula of {$\cos(2t)$} above!) {$\sin^2 t = \frac{1  \cos(2t)}{2}$}
 {$\cos^2 t = \frac{1 + \cos(2t)}{2}$}
 Do problems from the book for practice.