CiNotesPrecalculus formula sheet
A general precalculus sheet that fits most high school precalc classes, AP or not. Your class may skip some parts, and you can delete them in the editor.
Functions
| Domain | no dividing by 0, no even root of a negative, logs only of positive numbers |
|---|---|
| Composition | (f ∘ g)(x) = f(g(x)) |
| Inverse | swap x and y, solve for y; the graph reflects over y = x |
| Even and odd | even: f(−x) = f(x), symmetric over the y-axis; odd: f(−x) = −f(x), symmetric about the origin |
| Average rate of change | [f(b) − f(a)]/(b − a) |
Transformations of y = a·f(b(x − h)) + k
| Shifts | right h, up k |
|---|---|
| Vertical | |a| > 1 stretch; a < 0 flip over the x-axis |
| Horizontal | |b| > 1 compresses by a factor of 1/|b|; b < 0 flips over the y-axis |
Polynomials and rational functions
| Degree n | at most n real zeros and n − 1 turning points |
|---|---|
| End behavior | even degree: both ends the same way; odd degree: opposite ways; leading coefficient > 0 means the right end goes up |
| Multiplicity | odd: crosses the x-axis; even: touches and turns |
| Complex zeros | come in conjugate pairs a ± bi |
| Holes and vertical asymptotes | a factor that cancels gives a hole; a zero of the denominator only gives a vertical asymptote |
| Horizontal asymptote | top degree smaller: y = 0; equal: ratio of leading coefficients; top degree one bigger: slant asymptote |
Exponents and logarithms
| Exponential models | y = a·bˣ (growth if b > 1, decay if 0 < b < 1); y = a(1 ± r)t |
|---|---|
| Compound interest | A = P(1 + r/n)nt; continuous A = Pert |
| Half-life | y = a(½)t/h |
| Definition | logb x = y means bʸ = x |
| Log rules | log(xy) = log x + log y; log(x/y) = log x − log y; log(xᵖ) = p log x |
| Change of base | logb x = ln x / ln b |
Trigonometry
| Unit circle | point (cos t, sin t); sin2 t + cos2 t = 1; tan t = sin t / cos t |
|---|---|
| Sinusoid | y = A sin(B(x − C)) + D: amplitude |A|, period 2π/|B|, midline y = D |
| Tangent period | π/|B| |
| Degrees and radians | radians = degrees × π/180 |
Polar, parametric and vectors
| Polar to rectangular | x = r cos θ; y = r sin θ; r2 = x2 + y2; tan θ = y/x (check the quadrant) |
|---|---|
| Polar curves | r = a cos θ or a sin θ: circles; r = a + b cos θ: limaçon; r = a cos(nθ): rose |
| Parametric | x = f(t), y = g(t); eliminate t by solving one equation for t |
| Vector length | |⟨a, b⟩| = √(a2 + b2); unit vector v/|v| |
| Dot product | ⟨a, b⟩·⟨c, d⟩ = ac + bd; cos θ = (u·v)/(|u||v|); dot = 0 means perpendicular |
Conics
| Circle | (x − h)2 + (y − k)2 = r2 |
|---|---|
| Parabola | (x − h)2 = 4p(y − k), focus p units from the vertex |
| Ellipse | (x − h)2/a2 + (y − k)2/b2 = 1; c2 = a2 − b2 (a > b) |
| Hyperbola | (x − h)2/a2 − (y − k)2/b2 = 1; c2 = a2 + b2; asymptotes y − k = ±(b/a)(x − h) |
Sequences, series and matrices
| Arithmetic | an = a1 + (n − 1)d; Sn = n(a1 + an)/2 |
|---|---|
| Geometric | an = a1rn−1; Sn = a1(1 − rⁿ)/(1 − r) |
| Infinite geometric | S = a1/(1 − r) when |r| < 1 |
| 2×2 determinant | det [[a, b], [c, d]] = ad − bc |
| 2×2 inverse | (1/det)·[[d, −b], [−c, a]], none if det = 0; AX = B ⇒ X = A−1B |
On the TI-84 Plus CE
| Modes | MODE: RADIAN or DEGREE; POLAR or PAR for r= and x(t), y(t) graphs |
|---|---|
| Graph tools | 2ND TRACE (CALC): zero, minimum, maximum, intersect |
| Logs | LN key for natural log; MATH ▸ logBASE( for other bases |
| Matrices | 2ND x−1 (MATRIX) |
What it looks like on the calculator
The calculator version is the same sheet in plain text with short lines, so it fits the screen. Square roots are written as sqrt( ), powers with ^ and Greek letters as words. On the calculator, search a word like “asymptote” or “polar” to jump straight to it. The notes are stored in Archive, so a RAM clear won’t erase them.
Want your own class notes on there too?
Upload your teacher’s study guide or review packet as a PDF ($3 for up to 10 pages), or paste in your own notes. Diagrams come through as pictures.
Upload my notes or PDFCommon questions
How much does it cost to put this on my calculator?
This sheet is 655 words, so it’s a one-time $3. Typed notes are free up to 300 words and $3 from 301 to 10,000 words. If you cut it down to 300 words or fewer, it’s free. You can also add your own notes to the same order, up to 10,000 words in total (about 9,345 more words).
Is this the AP Precalculus formula sheet?
No. It’s a general precalculus sheet, not tied to one course outline. Functions, exponents and logs, trig and polar are in most precalc classes, including AP Precalculus. Conics, vectors and matrices depend on your class, so keep or delete them in the editor.
Can I change the notes before I pay?
Yes. The button opens the sheet in the CiNotes notes editor. Delete what you don’t need, add your teacher’s examples, and check the calculator preview before you check out.
Will it work on my calculator?
It works on the TI-84 Plus CE and the TI-84 Plus CE Python. It doesn’t work on the TI-84 Evo or on black-and-white TI-84s. If your calculator is on OS 5.5 or newer, it needs a free one-time setup first. The setup guide walks you through it.
Will the notes disappear if my calculator resets?
No. The notes are stored in Archive, so a RAM clear won’t erase them. On OS 5.5 or newer, open AsmHook2 from the APPS menu once after a RAM clear, then open your notes as usual.
More subject sheets: AP Chemistry, Chemistry, AP Physics 1, AP Calculus AB, AP Statistics, Unit circle and trig identities, Algebra 2. Or see all formula notes, every way to put notes on a TI-84, and what CiNotes is.