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.