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Precalculus Lippman Rasmussen

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Precalculus: An Investigation of Functions, by David Lippman and Melonie Rasmussen, licensed under CC BY-SA 4.0. Read the original free This shelf’s study material is shared under the same license (CC BY-SA 4.0).

Cheat sheet

Precalculus Cheat Sheet

Key terms

Term	Meaning
Function	Each input gets exactly one output
One-to-one	Each output comes from exactly one input
Domain / range	Allowed inputs / possible outputs
Linear function	$f(x)=b+mx$ : start value $b$ , constant rate $m$
Polynomial	$a_nx^n+\dots+a_0$ with whole-number powers ≥ 0
Degree / leading term	Highest power / the term with that power
Multiplicity	Power $p$ on a factor $(x-h)^p$
Exponential	$f(x)=ab^x$ , $b=1+r$ (growth factor)
Logarithm	$\log_b x$ is the exponent that turns $b$ into $x$
$e$	$\approx 2.718282$ , the limit of $(1+\frac1k)^k$
Radian	$\theta=s/r$ , arclength ÷ radius (no units)
Period	Smallest shift $P$ with $f(x+P)=f(x)$
Amplitude / midline	Height above center / center line $y=k$
Vector	Quantity with magnitude and direction, $\langle x,y\rangle$
Conic	Curve from a plane cutting a double cone
Ellipse	Distances to 2 foci have a constant sum
Hyperbola	Distances to 2 foci have a constant difference

Formulas and rules

Topic	Formula
Avg. rate / slope	$\frac{f(b)-f(a)}{b-a}$ , $m=\frac{y_2-y_1}{x_2-x_1}$
Perpendicular slopes	$m_2=-\frac{1}{m_1}$ (parallel: equal)
Vertex form	$a(x-h)^2+k$ ; $h=-\frac{b}{2a}$ , $k=f(h)$
Quadratic formula	$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factor theorem	$x-c$ is a factor $\iff p(c)=0$
Rational roots	candidates $\pm\frac{p}{q}$ : $p$ divides constant, $q$ divides leading coeff.
Compound interest	$A=a\left(1+\frac{r}{k}\right)^{kt}$
Log ↔ exponent	$b^a=c \iff \log_b c=a$
Log rules	$\log_b(AC)=\log_bA+\log_bC$ ; $\log_b\frac AC=\log_bA-\log_bC$ ; $\log_b A^r=r\log_bA$
Change of base	$\log_bA=\frac{\ln A}{\ln b}$
Distance / circle	$\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ ; $(x-h)^2+(y-k)^2=r^2$
Arclength	$s=r\theta$ ; $180^\circ=\pi$ rad
Unit circle	$(\cos\theta,\sin\theta)$ ; $\cos^2\theta+\sin^2\theta=1$
Sinusoid	$A\sin(Bt)+k$ , $P=\frac{2\pi}{B}$
Sum/difference	$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$ ; $\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$
Double angle	$\sin2\theta=2\sin\theta\cos\theta$ ; $\cos2\theta=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1$
Combine waves	$m\sin Bx+n\cos Bx=A\sin(Bx+C)$ , $A=\sqrt{m^2+n^2}$
Law of Sines	$\frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}$
Law of Cosines	$c^2=a^2+b^2-2ab\cos\gamma$
Polar	$x=r\cos\theta$ , $y=r\sin\theta$ , $r^2=x^2+y^2$
Euler / DeMoivre	$re^{i\theta}=r\cos\theta+ir\sin\theta$ ; $z^n=r^ne^{in\theta}$
Dot product	$a_1b_1+a_2b_2=
Ellipse	$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ , $c^2=a^2-b^2$
Hyperbola	$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$ , asymptotes $y-k=\pm\frac ba(x-h)$

Special angles:

$\theta$	$0$	$\frac\pi6$	$\frac\pi4$	$\frac\pi3$	$\frac\pi2$
$\cos$	1	$\frac{\sqrt3}{2}$	$\frac{\sqrt2}{2}$	$\frac12$	0
$\sin$	0	$\frac12$	$\frac{\sqrt2}{2}$	$\frac{\sqrt3}{2}$	1

Inverse trig ranges: $\sin^{-1}$ : $[-\frac\pi2,\frac\pi2]$ ; $\cos^{-1}$ : $[0,\pi]$ ; $\tan^{-1}$ : $(-\frac\pi2,\frac\pi2)$ .

Processes

Domain: exclude negative radicands and zero denominators. $\sqrt{2x+4}$ → $[-2,\infty)$ .
Line from 2 points: find $m$ , plug in one point, solve for $b$ .
Intersection / break-even: set formulas equal, solve for input, evaluate for output.
Polynomial from graph: zeros + multiplicity (cross = 1, bounce = 2, flat S = 3), write $a(x-x_1)^{p_1}\cdots$ , use one point to find $a$ .
Long division: divide leading terms, multiply, subtract; repeat until remainder degree < divisor degree.
Exponential equation: isolate $b^x$ , take log of both sides, bring exponent down, solve.
Sinusoid from graph: midline = (max+min)/2, amplitude = (max−min)/2, $B=2\pi/P$ ; cosine if starting at max/min, sine if on midline; negate if starting low.
Trig equation: use identities to get one function of one angle, factor, solve each factor on $[0,2\pi)$ .
Conic to standard form: group x and y terms, complete the square, divide so right side = 1.

Watch out

Wrong idea	Correction
Rational roots theorem finds all zeros	It only lists candidates to test; $x^2-2$ has zeros $\pm\sqrt2$
Divide both sides by $\sin x$	Loses $\sin x=0$ solutions; factor instead
Dot product gives a vector	It is a single number
$\cos^2\theta=\cos(\theta^2)$	It means $(\cos\theta)^2$
Bigger denominator sets hyperbola direction	The positive squared term does
$\tan^{-1}(y/x)$ is always the angle	Check quadrant from signs of $x,y$
Solving gives one answer	Evaluating gives one; solving may give several
Extrapolation is as reliable as interpolation	Outside the data it is less trustworthy

Mnemonics

"Vertical = Valid function, Horizontal = Honest one-to-one" — the two line tests.
"Log Gives the Exponent" — $\log_b c=a$ answers " $b$ to what power is $c$ ?"
"Cosine Contradicts" — cosine sum/difference flips the sign; sine keeps it.
"Ellipse Adds, Hyperbola Subtracts" — sum vs. difference of focal distances (and $c^2=a^2-b^2$ for ellipses).
"1-2-3 under the root, all over 2" — sines at $\frac\pi6,\frac\pi4,\frac\pi3$ are $\frac{\sqrt1}{2},\frac{\sqrt2}{2},\frac{\sqrt3}{2}$ .

Precalculus Lippman Rasmussen

Cheat sheet