Common templates. Click a question field first, then click a formula to insert editable LaTeX.
Math Structures Inline blank \( \) Fraction \frac{a}{b} \(\frac{a}{b}\) Blank fraction \frac{}{} \(\frac{}{}\) Mixed number 2\frac{1}{3} \(2\frac{1}{3}\) Square root \sqrt{x} \(\sqrt{x}\) n-th root \sqrt[n]{x} \(\sqrt[n]{x}\) Cube root \sqrt[3]{x} \(\sqrt[3]{x}\) n-th root with power \sqrt[n]{x^m} \(\sqrt[n]{x^m}\) Power x^{n} \(x^{n}\) Subscript a_n \(a_n\) Absolute value \left|x\right| \(\left|x\right|\) Large parentheses \left(x+1\right) \(\left(x+1\right)\) Ordered pair (x,y) \((x,y)\) Interval [a,b] \([a,b]\)
Common Equations Ratio \frac{a}{b}=\frac{c}{d} \(\frac{a}{b}=\frac{c}{d}\) Percent p\%=\frac{p}{100} \(p\%=\frac{p}{100}\) Average \text{average}=\frac{\text{sum}}{\text{number of terms}} \(\text{average}=\frac{\text{sum}}{\text{number of terms}}\) Distance-rate-time d=rt \(d=rt\) Simple interest I=Prt \(I=Prt\) Scientific notation a\times 10^n \(a\times 10^n\) Cases / system \left\{\begin{aligned} ax+by&=c\\ dx+ey&=f \end{aligned}\right. \[\left\{\begin{aligned} ax+by&=c\\ dx+ey&=f \end{aligned}\right.\] Aligned work \begin{aligned} 2x+3&=11\\ 2x&=8\\ x&=4 \end{aligned} \(\begin{aligned} 2x+3&=11\\ 2x&=8\\ x&=4 \end{aligned}\)
Algebra templates. Click a question field first, then click a formula to insert editable LaTeX.
Linear Equations, Inequalities, Exponents Slope-intercept y=mx+b \(y=mx+b\) Point-slope y-y_1=m(x-x_1) \(y-y_1=m(x-x_1)\) Standard form Ax+By=C \(Ax+By=C\) Slope formula m=\frac{y_2-y_1}{x_2-x_1} \(m=\frac{y_2-y_1}{x_2-x_1}\) Direct variation y=kx \(y=kx\) Inverse variation xy=k \(xy=k\) Compound inequality a<x<b \(a<x<b\) Inequality interval x\le a\text{ or }x\ge b \(x\le a\text{ or }x\ge b\) Exponent product x^a\cdot x^b=x^{a+b} \(x^a\cdot x^b=x^{a+b}\) Exponent quotient \frac{x^a}{x^b}=x^{a-b} \(\frac{x^a}{x^b}=x^{a-b}\) Power rule (x^a)^b=x^{ab} \((x^a)^b=x^{ab}\) Negative exponent x^{-n}=\frac{1}{x^n} \(x^{-n}=\frac{1}{x^n}\) Rational exponent x^{\frac{m}{n}}=\sqrt[n]{x^m} \(x^{\frac{m}{n}}=\sqrt[n]{x^m}\) Root-power relation \sqrt[n]{x^m}=x^{\frac{m}{n}} \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\)
Quadratics, Polynomials, Factoring Quadratic formula x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) Discriminant \Delta=b^2-4ac \(\Delta=b^2-4ac\) Vertex form y=a(x-h)^2+k \(y=a(x-h)^2+k\) Axis of symmetry x=\frac{-b}{2a} \(x=\frac{-b}{2a}\) Vertex \left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right) \(\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\) Difference squares a^2-b^2=(a-b)(a+b) \(a^2-b^2=(a-b)(a+b)\) Perfect square + (a+b)^2=a^2+2ab+b^2 \((a+b)^2=a^2+2ab+b^2\) Perfect square − (a-b)^2=a^2-2ab+b^2 \((a-b)^2=a^2-2ab+b^2\) Sum of cubes a^3+b^3=(a+b)(a^2-ab+b^2) \(a^3+b^3=(a+b)(a^2-ab+b^2)\) Difference cubes a^3-b^3=(a-b)(a^2+ab+b^2) \(a^3-b^3=(a-b)(a^2+ab+b^2)\) Factored polynomial f(x)=a(x-r_1)(x-r_2) \(f(x)=a(x-r_1)(x-r_2)\) Remainder theorem f(x)=(x-c)q(x)+f(c) \(f(x)=(x-c)q(x)+f(c)\)
Rational & Radical Expressions Rational equation \frac{a}{x+b}=c \(\frac{a}{x+b}=c\) Complex fraction \frac{\frac{a}{b}}{\frac{c}{d}} \(\frac{\frac{a}{b}}{\frac{c}{d}}\) Conjugates (a+\sqrt{b})(a-\sqrt{b})=a^2-b \((a+\sqrt{b})(a-\sqrt{b})=a^2-b\) Radical product \sqrt{ab}=\sqrt a\sqrt b \(\sqrt{ab}=\sqrt a\sqrt b\) Radical quotient \sqrt{\frac{a}{b}}=\frac{\sqrt a}{\sqrt b} \(\sqrt{\frac{a}{b}}=\frac{\sqrt a}{\sqrt b}\) Rationalize \frac{1}{\sqrt a}\cdot\frac{\sqrt a}{\sqrt a}=\frac{\sqrt a}{a} \(\frac{1}{\sqrt a}\cdot\frac{\sqrt a}{\sqrt a}=\frac{\sqrt a}{a}\)
Functions / Logs templates. Click a question field first, then click a formula to insert editable LaTeX.
Function Notation and Transformations Function f(x)= \(f(x)=\) Composition (f\circ g)(x)=f(g(x)) \((f\circ g)(x)=f(g(x))\) Inverse function f^{-1}(x) \(f^{-1}(x)\) Domain set \{x\mid x\ge 0\} \(\{x\mid x\ge 0\}\) Piecewise f(x)=\begin{cases} x, & x\ge 0\\ -x, & x<0 \end{cases} \(f(x)=\begin{cases} x, & x\ge 0\\ -x, & x<0 \end{cases}\) Transform right f(x-h)+k \(f(x-h)+k\) Average rate \frac{f(b)-f(a)}{b-a} \(\frac{f(b)-f(a)}{b-a}\) Even test f(-x)=f(x) \(f(-x)=f(x)\) Odd test f(-x)=-f(x) \(f(-x)=-f(x)\)
Logarithms and Exponentials Log \log(x) \(\log(x)\) Natural log \ln(x) \(\ln(x)\) Log base \log_b(x) \(\log_b(x)\) Log definition b^x=y\Longleftrightarrow \log_b(y)=x \(b^x=y\Longleftrightarrow \log_b(y)=x\) Product law \log_b(MN)=\log_b M+\log_b N \(\log_b(MN)=\log_b M+\log_b N\) Quotient law \log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N \(\log_b\left(\frac{M}{N}\right)=\log_b M-\log_b N\) Power law \log_b(M^p)=p\log_b M \(\log_b(M^p)=p\log_b M\) Change of base \log_b a=\frac{\log a}{\log b} \(\log_b a=\frac{\log a}{\log b}\) Exponential y=ab^x \(y=ab^x\) Growth / decay y=a(1+r)^t \(y=a(1+r)^t\) Continuous growth A=Pe^{rt} \(A=Pe^{rt}\)
Sequences and Series Arithmetic term a_n=a_1+(n-1)d \(a_n=a_1+(n-1)d\) Arithmetic sum S_n=\frac{n(a_1+a_n)}{2} \(S_n=\frac{n(a_1+a_n)}{2}\) Geometric term a_n=a_1r^{n-1} \(a_n=a_1r^{n-1}\) Finite geometric sum S_n=\frac{a_1(1-r^n)}{1-r} \(S_n=\frac{a_1(1-r^n)}{1-r}\) Infinite geometric sum S_\infty=\frac{a_1}{1-r},\ |r|<1 \(S_\infty=\frac{a_1}{1-r},\ |r|<1\) Sigma notation \sum_{k=1}^{n} a_k \(\sum_{k=1}^{n} a_k\) Sigma formula \sum_{k=1}^{n} k=\frac{n(n+1)}{2} \(\sum_{k=1}^{n} k=\frac{n(n+1)}{2}\) Binomial theorem (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \((a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)
Geometry / Trig templates. Click a question field first, then click a formula to insert editable LaTeX.
Coordinate Geometry Distance formula d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) Midpoint M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \(M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\) Circle equation (x-h)^2+(y-k)^2=r^2 \((x-h)^2+(y-k)^2=r^2\) Parabola focus (x-h)^2=4p(y-k) \((x-h)^2=4p(y-k)\) Ellipse \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1 \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\) Hyperbola \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
Geometry Notation and Figures Segment \overline{AB} \(\overline{AB}\) Ray \overrightarrow{AB} \(\overrightarrow{AB}\) Line \overleftrightarrow{AB} \(\overleftrightarrow{AB}\) Angle \angle ABC \(\angle ABC\) Triangle \triangle ABC \(\triangle ABC\) Arc \overset{\frown}{AB} \(\overset{\frown}{AB}\) Parallel AB\parallel CD \(AB\parallel CD\) Perpendicular AB\perp CD \(AB\perp CD\) Congruent \triangle ABC\cong\triangle DEF \(\triangle ABC\cong\triangle DEF\) Similar \triangle ABC\sim\triangle DEF \(\triangle ABC\sim\triangle DEF\)
Plane Geometry Formulas Triangle area A=\frac{1}{2}bh \(A=\frac{1}{2}bh\) Rectangle area A=lw \(A=lw\) Circle area A=\pi r^2 \(A=\pi r^2\) Circumference C=2\pi r \(C=2\pi r\) Sector area A=\frac{\theta}{360^\circ}\pi r^2 \(A=\frac{\theta}{360^\circ}\pi r^2\) Arc length s=r\theta \(s=r\theta\) Interior polygon sum S=(n-2)180^\circ \(S=(n-2)180^\circ\) Pythagorean theorem a^2+b^2=c^2 \(a^2+b^2=c^2\) 45-45-90 x,x,x\sqrt2 \(x,x,x\sqrt2\) 30-60-90 x,x\sqrt3,2x \(x,x\sqrt3,2x\) Similarity ratio \frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF} \(\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}\)
Trigonometry Sine ratio \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) Cosine ratio \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) Tangent ratio \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) Inverse sine \sin^{-1}(x) \(\sin^{-1}(x)\) Pythagorean identity \sin^2\theta+\cos^2\theta=1 \(\sin^2\theta+\cos^2\theta=1\) Tangent identity \tan\theta=\frac{\sin\theta}{\cos\theta} \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) Law of sines \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) Law of cosines c^2=a^2+b^2-2ab\cos C \(c^2=a^2+b^2-2ab\cos C\) Radians 180^\circ=\pi\text{ radians} \(180^\circ=\pi\text{ radians}\) Unit circle point (\cos\theta,\sin\theta) \((\cos\theta,\sin\theta)\)
Statistics / Probability templates. Click a question field first, then click a formula to insert editable LaTeX.
Statistics and Data Mean \bar{x}=\frac{\sum x_i}{n} \(\bar{x}=\frac{\sum x_i}{n}\) Weighted mean \bar{x}=\frac{\sum w_ix_i}{\sum w_i} \(\bar{x}=\frac{\sum w_ix_i}{\sum w_i}\) Population SD \sigma=\sqrt{\frac{\sum (x_i-\mu)^2}{N}} \(\sigma=\sqrt{\frac{\sum (x_i-\mu)^2}{N}}\) Sample SD s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}} \(s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}\) z-score z=\frac{x-\mu}{\sigma} \(z=\frac{x-\mu}{\sigma}\) IQR IQR=Q_3-Q_1 \(IQR=Q_3-Q_1\) Line of best fit \hat{y}=a+bx \(\hat{y}=a+bx\) Correlation r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2\sum (y_i-\bar{y})^2}} \(r=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2\sum (y_i-\bar{y})^2}}\)
Probability and Counting Probability P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}} \(P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}\) Complement P(A^c)=1-P(A) \(P(A^c)=1-P(A)\) Conditional probability P(A\mid B)=\frac{P(A\cap B)}{P(B)} \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\) Addition rule P(A\cup B)=P(A)+P(B)-P(A\cap B) \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) Independent events P(A\cap B)=P(A)P(B) \(P(A\cap B)=P(A)P(B)\) Factorial n! \(n!\) Permutation {}_nP_r=\frac{n!}{(n-r)!} \({}_nP_r=\frac{n!}{(n-r)!}\) Combination {}_nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!} \({}_nC_r=\binom{n}{r}=\frac{n!}{r!(n-r)!}\) Expected value E(X)=\sum x_iP(x_i) \(E(X)=\sum x_iP(x_i)\) Binomial probability P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
Matrices / Vectors templates. Click a question field first, then click a formula to insert editable LaTeX.
Matrices 2×2 matrix \begin{bmatrix}a&b\\c&d\end{bmatrix} \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\) 3×3 matrix \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} \(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\) Identity 2×2 I=\begin{bmatrix}1&0\\0&1\end{bmatrix} \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) Determinant 2×2 \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\) Matrix equation A\vec{x}=\vec{b} \(A\vec{x}=\vec{b}\) Augmented matrix \left[\begin{array}{cc|c}a&b&e\\c&d&f\end{array}\right] \(\left[\begin{array}{cc|c}a&b&e\\c&d&f\end{array}\right]\)
Vectors and Complex Numbers Vector \vec{v}=\langle a,b\rangle \(\vec{v}=\langle a,b\rangle\) Magnitude \left\|\vec{v}\right\|=\sqrt{a^2+b^2} \(\left\|\vec{v}\right\|=\sqrt{a^2+b^2}\) Dot product \vec{u}\cdot\vec{v}=u_1v_1+u_2v_2 \(\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2\) Complex number a+bi \(a+bi\) Complex conjugate a-bi \(a-bi\) Polar complex z=r(\cos\theta+i\sin\theta) \(z=r(\cos\theta+i\sin\theta)\)
Sets / Symbols templates. Click a question field first, then click a formula to insert editable LaTeX.
Relations and Logic Less/equal \le \(\le\) Greater/equal \ge \(\ge\) Not equal \ne \(\ne\) Approximately \approx \(\approx\) Plus/minus \pm \(\pm\) Infinity \infty \(\infty\) Implies \Rightarrow \(\Rightarrow\) If and only if \Longleftrightarrow \(\Longleftrightarrow\) Therefore \therefore \(\therefore\) Because \because \(\because\)
Sets and Number Systems Element x\in A \(x\in A\) Not element x\notin A \(x\notin A\) Union A\cup B \(A\cup B\) Intersection A\cap B \(A\cap B\) Subset A\subseteq B \(A\subseteq B\) Proper subset A\subset B \(A\subset B\) Empty set \varnothing \(\varnothing\) Complement A^c \(A^c\) Real numbers \mathbb{R} \(\mathbb{R}\) Integers \mathbb{Z} \(\mathbb{Z}\) Rationals \mathbb{Q} \(\mathbb{Q}\) Naturals \mathbb{N} \(\mathbb{N}\)
Greek Letters Theta \theta \(\theta\) Alpha \alpha \(\alpha\) Beta \beta \(\beta\) Gamma \gamma \(\gamma\) Delta \Delta \(\Delta\) Pi \pi \(\pi\) Mu \mu \(\mu\) Sigma \sigma \(\sigma\) Omega \Omega \(\Omega\)
Derivs / AP templates. Click a question field first, then click a formula to insert editable LaTeX.
Limits and Calculus Readiness Limit \lim_{x\to a} f(x) \(\lim_{x\to a} f(x)\) Limit at infinity \lim_{x\to\infty} f(x) \(\lim_{x\to\infty} f(x)\) Derivative definition f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h} \(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\) Derivative \frac{d}{dx}f(x) \(\frac{d}{dx}f(x)\) Second derivative \frac{d^2y}{dx^2} \(\frac{d^2y}{dx^2}\) Integral \int f(x)\,dx \(\int f(x)\,dx\) Definite integral \int_a^b f(x)\,dx \(\int_a^b f(x)\,dx\) Sigma as area \lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x \(\lim_{n\to\infty}\sum_{i=1}^n f(x_i)\Delta x\)
AP Precalculus / Advanced Functions Rational function f(x)=\frac{p(x)}{q(x)} \(f(x)=\frac{p(x)}{q(x)}\) Asymptote x=a\text{ is a vertical asymptote} \(x=a\text{ is a vertical asymptote}\) Polar point (r,\theta) \((r,\theta)\) Parametric \left\{\begin{array}{l}x=f(t)\\y=g(t)\end{array}\right. \(\left\{\begin{array}{l}x=f(t)\\y=g(t)\end{array}\right.\) Conic general Ax^2+Bxy+Cy^2+Dx+Ey+F=0 \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\)