AP Precalculus Formula Sheet

What You Need to Know

AP Precalculus problems are usually function problems: you’re given an equation/graph/table/context, and you must analyze features (domain/range, intercepts, end behavior, asymptotes, period, growth rate) and build or solve models (polynomial/rational, exponential/logarithmic, trigonometric, sequences/series). This sheet is the high-yield formulas + procedures you’ll actually reach for under time pressure.

Critical reminder: Every time you do algebra on a model, track domain restrictions (especially for rational and logarithmic functions) and units/mode (degrees vs radians) for trig.

Core “language of functions” you must be fluent in

Function notation: $f(x)$ , inputs $x$ , outputs $y$ .
Average rate of change on $[a,b]$ (slope of secant line):
$\frac{f(b)-f(a)}{b-a}$
Composition: $(f\circ g)(x)=f(g(x))$
Inverse: $f^{-1}(x)$ undoes $f(x)$ , so
$f(f^{-1}(x))=x \text{ and } f^{-1}(f(x))=x$
(on appropriate domains).

Step-by-Step Breakdown

1) Analyze a rational function fast

Given $f(x)=\frac{p(x)}{q(x)}$ :

Factor $p(x)$ and $q(x)$ completely.
Cancel common factors (if any).
- If a factor cancels, you have a hole at that $x$ -value.
Domain: exclude values making the original denominator $0$ .
Vertical asymptotes (VA): zeros of the simplified denominator.
Horizontal/oblique asymptote: compare degrees.
- If $\deg(p)<\deg(q)$ , HA is $y=0$ .
- If $\deg(p)=\deg(q)$ , HA is ratio of leading coefficients.
- If $\deg(p)=\deg(q)+1$ , do division for a slant asymptote.
Intercepts:
- $x$ -intercepts: zeros of simplified numerator (that are in domain).
- $y$ -intercept: evaluate $f(0)$ if defined.

2) Solve exponential/log equations cleanly

Isolate the exponential or logarithm.
Use:
- If you have $b^{\text{stuff}}=b^{\text{stuff}}$ , set exponents equal.
- If bases differ, take logs: $\ln(\cdot)$ or $\log(\cdot)$ .
Check for extraneous solutions when logs are involved (arguments must be positive).

Mini-example: Solve $3\cdot 2^{x-1}=24$

Divide by $3$ : $2^{x-1}=8=2^3$
Exponents: $x-1=3 \Rightarrow x=4$

3) Build a sinusoidal model from features

For periodic data/graphs, use:
$y=A\sin(B(x-C))+D \quad \text{or} \quad y=A\cos(B(x-C))+D$
Steps:

Midline: $D=\frac{\text{max}+\text{min}}{2}$
Amplitude: $|A|=\frac{\text{max}-\text{min}}{2}$
Period: $P=\text{horizontal length of one cycle}$ , then $B=\frac{2\pi}{P}$ (sine/cosine).
Phase shift: choose $C$ so the curve hits a key point (cos starts at a peak; sin starts at midline increasing).

4) Decide arithmetic vs geometric sequences quickly

Given a sequence $a_1,a_2,a_3,\dots$ :

Compute differences: $a_{n}-a_{n-1}$ .
- Constant difference $\Rightarrow$ arithmetic.
Compute ratios: $\frac{a_n}{a_{n-1}}$ (when terms nonzero).
- Constant ratio $\Rightarrow$ geometric.
Use explicit formula to jump to $a_n$ ; use sum formulas for totals.

Key Formulas, Rules & Facts

A) Transformations & graph moves

Form	What it does	Notes
$y=f(x)+k$	vertical shift	up if $k>0$
$y=f(x-k)$	horizontal shift	right if $k>0$ (opposite sign!)
$y=af(x)$	vertical stretch/shrink + reflect	reflect over $x$ -axis if $a<0$
$y=f(ax)$	horizontal shrink/stretch + reflect	shrink by $\frac{1}{\|a\|}$ ; reflect over $y$ -axis if $a<0$
$y=\|f(x)\|$	reflect negative outputs up	affects $y$ values
$y=f(\|x\|)$	mirror right half to left	affects $x$ values

B) Polynomial essentials

Rule	Formula	Use
End behavior	leading term dominates	determine left/right tails
Factor Theorem	$f(c)=0 \Leftrightarrow (x-c) \text{ is a factor}$	connect zeros to factors
Remainder Theorem	remainder of division by $(x-c)$ is $f(c)$	fast remainder
Multiplicity	factor $(x-c)^m$	odd $m$ crosses; even $m$ touches/bounces
Complex zeros	if coefficients real and $a+bi$ is a zero, then $a-bi$ is a zero	conjugate pairs

C) Rational functions

Feature	How to find	Notes
Domain	exclude zeros of original denominator	even if a factor cancels
Hole	common factor cancels	point is missing, not an asymptote
Vertical asymptote	zero of simplified denominator	graph shoots to $\pm\infty$
Horizontal asymptote	degree comparison	see rules below
Slant asymptote	polynomial division when degree differs by 1	asymptote is a line

Horizontal asymptote rules for $\frac{p(x)}{q(x)}$ :

If $\deg(p)<\deg(q)$ , then $y=0$ .
If $\deg(p)=\deg(q)$ , then $y=\frac{\text{lead coeff of }p}{\text{lead coeff of }q}$ .
If $\deg(p)>\deg(q)$ , no horizontal asymptote (possible slant/curved asymptote via division).

D) Exponentials & logarithms

Concept	Formula	Notes
Exponential model	$y=a\cdot b^x$	$a$ is initial value at $x=0$
Growth/decay (percent)	$y=a(1\pm r)^t$	$r$ as decimal, $t$ in time units
Continuous growth/decay	$y=ae^{kt}$	$k>0$ grow, $k<0$ decay
Log definition	$\log_b(x)=y \Leftrightarrow b^y=x$	requires $b>0$ , $b\ne 1$ , $x>0$
Product	$\log_b(MN)=\log_b(M)+\log_b(N)$	expand/condense logs
Quotient	$\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$	arguments positive
Power	$\log_b(M^p)=p\log_b(M)$	the only legal way to “move” exponents
Change of base	$\log_b(x)=\frac{\ln(x)}{\ln(b)}=\frac{\log(x)}{\log(b)}$	use calculator with $\ln$ or $\log$

Compounding interest (common forms):

Compounded $n$ times per year:
$A=P\left(1+\frac{r}{n}\right)^{nt}$
Continuous compounding:
$A=Pe^{rt}$

E) Trigonometry (unit-circle level)

Item	Formula	Notes
Degrees ↔ radians	$\pi \text{ rad}=180^\circ$	multiply by $\frac{\pi}{180}$ or $\frac{180}{\pi}$
Pythagorean identity	$\sin^2(\theta)+\cos^2(\theta)=1$	from unit circle
Tangent	$\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$	undefined where $\cos(\theta)=0$
Sec/csc/cot	$\sec=\frac{1}{\cos}$ , $\csc=\frac{1}{\sin}$ , $\cot=\frac{\cos}{\sin}$	reciprocal identities
Even/odd	$\cos(-\theta)=\cos(\theta)$ ; $\sin(-\theta)=-\sin(\theta)$	symmetry
Periodicity	$\sin(\theta+2\pi)=\sin(\theta)$ ; $\cos(\theta+2\pi)=\cos(\theta)$	wrap around

Special angles (radians):

$\theta$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$
$\sin(\theta)$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos(\theta)$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan(\theta)$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	undefined

F) Sinusoidal functions (graphs & parameters)

Function	Standard form	Key facts
Sine	$y=A\sin(B(x-C))+D$	amplitude $\|A\|$ ; period $\frac{2\pi}{\|B\|}$
Cosine	$y=A\cos(B(x-C))+D$	same amplitude/period rules
Tangent	$y=A\tan(B(x-C))+D$	period $\frac{\pi}{\|B\|}$ ; VAs each half-period

Midline & range:

Midline: $y=D$
Range for sine/cosine: $[D-|A|,\,D+|A|]$

G) Triangle trig (when you’re not on the unit circle)

Tool	Formula	When to use
Law of Sines	$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$	AAS, ASA, or SSA (ambiguous case!)
Law of Cosines	$c^2=a^2+b^2-2ab\cos(C)$	SAS or SSS
Triangle area	$K=\frac{1}{2}ab\sin(C)$	two sides + included angle

SSA warning: Law of Sines can yield 0, 1, or 2 triangles depending on side/angle sizes.

H) Sequences & series

Type	Explicit	Recursive	Sum
Arithmetic	$a_n=a_1+(n-1)d$	$a_n=a_{n-1}+d$	$S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)$
Geometric	$a_n=a_1r^{n-1}$	$a_n=ra_{n-1}$	$S_n=a_1\frac{1-r^n}{1-r}$ for $r\ne 1$

Infinite geometric series: if $|r|<1$ ,
$S_\infty=\frac{a_1}{1-r}$

Examples & Applications

1) Rational features (hole vs asymptote)

Analyze $f(x)=\frac{(x-2)(x+1)}{(x-2)(x-3)}$ .

Cancel $(x-2)$ : simplified $f(x)=\frac{x+1}{x-3}$ , but $x\ne 2$ .
Hole at $x=2$ (not a vertical asymptote). Hole’s $y$ -value from simplified form:
$f(2)=\frac{2+1}{2-3}=-3$ , so hole at $(2,-3)$ .
VA: $x=3$ .
Degrees equal, HA is ratio of leads: $y=1$ .

2) Log equation with restriction check

Solve $\ln(x-1)=\ln(5)$ .

Set arguments equal: $x-1=5 \Rightarrow x=6$ .
Check domain: need $x-1>0 \Rightarrow x>1$ , so $x=6$ works.

3) Sinusoid from max/min/period

A function has max $8$ , min $2$ , and period $12$ . Write a cosine model with no phase shift.

Midline: $D=\frac{8+2}{2}=5$
Amplitude: $|A|=\frac{8-2}{2}=3$
Period: $P=12 \Rightarrow B=\frac{2\pi}{12}=\frac{\pi}{6}$
Cosine with peak at $x=0$ : $y=3\cos\left(\frac{\pi}{6}x\right)+5$

4) Infinite geometric sum (modeling repeating patterns)

Find $0.7+0.07+0.007+\cdots$

Geometric with $a_1=0.7$ and $r=0.1$ .
Since $|r|<1$ ,
$S_\infty=\frac{0.7}{1-0.1}=\frac{0.7}{0.9}=\frac{7}{9}$

Common Mistakes & Traps

Canceling factors and forgetting the hole
- Wrong: cancel $(x-2)$ and then treat $x=2$ as allowed.
- Fix: if it canceled, $x=2$ is still excluded; record a hole.
Calling every denominator zero a vertical asymptote
- Wrong: zero from a factor that cancels.
- Fix: VAs come from zeros of the simplified denominator; canceled zeros become holes.
Solving log equations without domain checks
- Wrong: solve algebra, keep a solution with $x\le 1$ in $\ln(x-1)$ .
- Fix: enforce $\text{argument}>0$ before/after solving.
Mixing degrees and radians (or wrong calculator mode)
- Wrong: using $\sin(90)$ expecting $1$ while in radian mode.
- Fix: confirm mode; remember $\frac{\pi}{2}$ corresponds to $90^\circ$ .
Getting sinusoid period backwards
- Wrong: period of $\sin(Bx)$ as $2\pi B$ .
- Fix: period is $\frac{2\pi}{|B|}$ (and $\frac{\pi}{|B|}$ for tangent).
Forgetting that inside transformations act “opposite”
- Wrong: $f(x-3)$ shifts left.
- Fix: $f(x-3)$ shifts **right** $3$ .
Assuming SSA always gives one triangle
- Wrong: apply Law of Sines and stop.
- Fix: SSA can be ambiguous; check if a second angle $180^\circ-\theta$ is possible.
Misusing inverse notation
- Wrong: interpreting $f^{-1}(x)$ as $\frac{1}{f(x)}$ .
- Fix: $f^{-1}$ is the **inverse function** (undoes $f$ ), not a reciprocal.

Memory Aids & Quick Tricks

Trick / mnemonic	What it helps you remember	When to use
“Inside is opposite”	$f(x-k)$ shifts right; $f(kx)$ compresses	transformations
SOH–CAH–TOA	$\sin=\frac{\text{opp}}{\text{hyp}}$ , $\cos=\frac{\text{adj}}{\text{hyp}}$ , $\tan=\frac{\text{opp}}{\text{adj}}$	right-triangle trig
ASTC (“All Students Take Calculus”)	signs of trig by quadrant	unit circle sign decisions
“Degree compare”	rational end behavior/asymptotes depend on degrees	rational functions
“Log arguments must be positive”	domain restrictions for $\log$ and $\ln$	all log solving
“Midline is average, amplitude is half-diff”	$D=\frac{\text{max}+\text{min}}{2}$ , $\|A\|=\frac{\text{max}-\text{min}}{2}$	sinusoidal modeling

Quick Review Checklist

You can compute average rate of change: $\frac{f(b)-f(a)}{b-a}$ .
You can do transformations and remember inside is opposite.
For rational functions, you can quickly identify domain, holes, VAs, HAs/slants, intercepts.
You know HA rules via degree comparison.
You can solve exponentials/logs and check log domains.
You know log properties and change of base: $\log_b(x)=\frac{\ln(x)}{\ln(b)}$ .
You can convert degrees/radians using $\pi \leftrightarrow 180^\circ$ .
You can model periodic behavior with $y=A\sin(B(x-C))+D$ and get $A,B,C,D$ from features.
You know arithmetic/geometric explicit + sums, and infinite geometric condition $|r|<1$ .

You’ve got this—use this sheet to set up problems fast, then let your algebra do the rest.