Precalculus Summary
Functions to know
- linears, quadratics, polynomials, rationals
- exponentials, 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
Quadratics
- {$f(x) = ax^2 + bx + c$} ↔ {$f(x) = a(x-h)^2 + k$} (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 {$\frac{P(x)}{D(x)}$}
- 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 {$(x-c)$} 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 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.
- {$a\sin(k(x-b))$}, {$a\cos(k(x-b))$}, {$a\tan(k(x-b))$}
- 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))$} – draw a triangle and figure out the lengths
Trigonometric algebra
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Be able to simplify expressions, prove identities, and solve equations.
-
Identities (know the first column of formulas)
| Important ones | Related ones | |
|---|---|---|
| Pythagorean | {$\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.