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)
- 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 |
| y=f(ax) | horizontal shrink/stretch + reflect | shrink by \frac{1}{|a|}; reflect over y-axis if a |
| 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)
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 |
| 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|
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|
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|
You’ve got this—use this sheet to set up problems fast, then let your algebra do the rest.