Precalculus Summary
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.