# Precalculus Summary

## Functions to know

• 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: {$f(x)$} → {$c f(\frac{x-a}{b}) + d$}
• 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

• {$\sqrt{4-x^2}$}, {$\sqrt{x^2-4}$} – can do directly or use sign analysis
• {$\frac{x+4}{x^2-4}$}, {$\frac{1}{\sqrt{x^2-4}}$}, {$\log(1-x^2)$}, {$e^\frac{1}{1-x}$}
• {$\sqrt{\frac{x+4}{x^2-4}}$} – requires sign analysis of rational function!
• {$\ln\left(\frac{x+4}{x^2-4}\right)$} – requires sign analysis of rational function!
• Compare to the domain of {$\ln(x+4) - \ln(x^2-4)$}
• Domain of {$f \circ g$} where {$f(x) = \frac{1}{x}$} and {$g(x) = \frac{1}{x-1}$}. Note that {$(f \circ g)(x) = x-1$}, but the domain of {$f \circ g$} does not include {$x=1$} since it is not in the domain of {$g$}.

### One-to-one 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 {$\sqrt[3]{x^5-1}$}, {$\sqrt[3]{\arcsin(x)-1}$}, {$\sqrt[3]{e^x-1}$}, {$\sqrt[3]{\log_5 x-1}$}
• Inverse of {$\frac{x+1}{x-1}$}, {$\log\left(\frac{x+1}{x-1}\right)$}, {$\arctan\left(\frac{x+1}{x-1}\right)$}

## 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

• {$f(x) = ax^2 + bx + c$}         ↔         {$f(x) = a(x-h)^2 + k$} (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: {$x^2 + (2d)x = x^2 + (2d)x + d^2 - d^2 = (x+d)^2-d^2$}
• 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 {$n$}: {$f(x)=a_n x^n + a_{n-1} x^{n-1} + … + a_{1}x + a_{0}$}, {$a_n \ne 0$}

• 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. {$f(c)=0$}
• {$x-c$} 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: {$P(x)=a_n x^n + a_{n-1} x^{n-1} + … + a_{1}x + a_{0}$}, {$a_n \ne 0$}
• 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 {$D(x) = x-c$}, 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 {$\frac{P(x)}{D(x)}$}
• Synthetic division can be used to quickly divide {$P$} by {$(x-c)$} to simultaneously test if {$c$} is a factor and to obtain a factorization.

### Rational functions

To analyze, {$f(x) = \frac{(x-1)(2x+3)(x-2)}{(x-3)^2(x-1)}$}

• Cancel out holes first: {$f(x) = \frac{(2x+3)(x-2)}{(x-3)^2}\cdot\frac{x-1}{x-1} = \frac{(2x+3)(x-2)}{(x-3)^2}$}
• 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 {$\frac{a^x}{a^y} = a^{x-y}$} {$\log_a (\frac{X}{Y}) = \log_a X - \log_a Y$}
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): {$T(t) = T_s + (T_0-T_s)e^{-kt}$}.
• 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 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.
• {$a\sin(k(x-b))$}, {$a\cos(k(x-b))$}, {$a\tan(k(x-b))$}
• 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 {$\sec, \csc, \cot$}
Co-function {$\cos\left(\frac{\pi}{2}-t\right) = \sin t$}
{$\sin\left(\frac{\pi}{2}-t\right) = \cos t$}
similarly for {$\sec/\csc, \tan/\cot$}
Angle sum {$\cos(s+t) = \cos s \cos t - \sin s \sin t$}
{$\sin(s+t) = \sin s \cos t + \cos s \sin t$}
{$\cos(s-t) = \cos s \cos t + \sin s \sin t$}
{$\sin(s-t) = \sin s \cos t - \cos s \sin t$}
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.