Formulas and reference

Absolute value of a number

NMT

\[|a|=\begin{cases}a,&a\ge0\\-a,&a\lt 0\end{cases}\]

6–9 grade

Absolute value of a product

\[|ab|=|a||b|\]

7–9 grade

Absolute value of a quotient

\[\left|\frac ab\right|=\frac{|a|}{|b|}\]

Condition \(b\ne0\)

7–9 grade

Fundamental property of a proportion

\[\frac ab=\frac cd\Longleftrightarrow ad=bc\]

Condition \(b\ne0,\ d\ne0\)

6–8 grade

p percent of a number

\[B=A\cdot\frac{p}{100}\]

5–9 grade

What percent B is of A

\[p=\frac BA\cdot100\%\]

Condition \(A\ne0\)

5–9 grade

Changing a number by p percent

\[A_{\uparrow}=A\left(1+\frac p{100}\right)\]

\[A_{\downarrow}=A\left(1-\frac p{100}\right)\]

6–9 grade

Direct proportion

\[y=kx\]

Condition \(k\ne0\)

6–9 grade

Inverse proportion

\[y=\frac{k}{x}\]

Condition \(k\ne0,\ x\ne0\)

6–9 grade

Zero exponent

NMT

\[a^0=1\]

Condition \(a\ne0\)

7–10 grade

Negative integer exponent

NMT

\[a^{-n}=\frac1{a^n}\]

Condition \(a\ne0,\ n\in\mathbb N\)

8–10 grade

Product of powers

NMT

\[a^m a^n=a^{m+n}\]

7–10 grade

Quotient of powers

NMT

\[\frac{a^m}{a^n}=a^{m-n}\]

Condition \(a\ne0\)

7–10 grade

Power of a power

NMT

\[(a^m)^n=a^{mn}\]

7–10 grade

Power of a product

NMT

\[(ab)^n=a^n b^n\]

7–10 grade

Power of a quotient

NMT

\[\left(\frac ab\right)^n=\frac{a^n}{b^n}\]

Condition \(b\ne0\)

7–10 grade

Rational exponent

NMT

\[a^{\frac mn}=\sqrt[n]{a^m}\]

Condition \(a\gt 0,\ m\in\mathbb Z,\ n\in\mathbb N,\ n\ge2\)

9–11 grade

Square root of a square

NMT

\[\sqrt{a^2}=|a|\]

8–10 grade

Root of a product

\[\sqrt{ab}=\sqrt a\sqrt b\]

Condition \(a\ge0,\ b\ge0\)

8–10 grade

Root of a quotient

\[\sqrt{\frac ab}=\frac{\sqrt a}{\sqrt b}\]

Condition \(a\ge0,\ b\gt 0\)

8–10 grade

Difference of squares

NMT

\[a^2-b^2=(a-b)(a+b)\]

7–9 grade

Square of a sum

NMT

\[(a+b)^2=a^2+2ab+b^2\]

7–9 grade

Square of a difference

NMT

\[(a-b)^2=a^2-2ab+b^2\]

7–9 grade

Cube of a sum

\[(a+b)^3=a^3+3a^2b+3ab^2+b^3\]

7–9 grade

Cube of a difference

\[(a-b)^3=a^3-3a^2b+3ab^2-b^3\]

7–9 grade

Sum of cubes

\[a^3+b^3=(a+b)(a^2-ab+b^2)\]

7–9 grade

Difference of cubes

\[a^3-b^3=(a-b)(a^2+ab+b^2)\]

7–9 grade

Adding algebraic fractions

\[\frac ab+\frac cd=\frac{ad+bc}{bd}\]

Condition \(b\ne0,\ d\ne0\)

8–9 grade

Multiplying algebraic fractions

\[\frac ab\cdot\frac cd=\frac{ac}{bd}\]

Condition \(b\ne0,\ d\ne0\)

8–9 grade

Dividing algebraic fractions

\[\frac ab:\frac cd=\frac{ad}{bc}\]

Condition \(b\ne0,\ c\ne0,\ d\ne0\)

8–9 grade

Linear equation

\[ax+b=0\Rightarrow x=-\frac ba\]

Condition \(a\ne0\)

7–9 grade

Equation with absolute value

\[|f(x)|=a\Longleftrightarrow f(x)=\pm a\]

Condition \(a\ge0\)

8–11 grade

Basic radical equation

\[\sqrt{f(x)}=g(x)\Longleftrightarrow\begin{cases}f(x)=g^2(x)\\g(x)\ge0\end{cases}\]

9–11 grade

Exponential equation with equal bases

\[a^{f(x)}=a^{g(x)}\Longleftrightarrow f(x)=g(x)\]

Condition \(a\gt 0,\ a\ne1\)

10–11 grade

Logarithmic equation with equal bases

\[\log_a f(x)=\log_a g(x)\Longleftrightarrow\begin{cases}f(x)=g(x)\\f(x)\gt 0\\g(x)\gt 0\end{cases}\]

Condition \(a\gt 0,\ a\ne1\)

10–11 grade

Quadratic equation

NMT

\[ax^2+bx+c=0\]

Condition \(a\ne0\)

8–11 grade

Discriminant

NMT

\[D=b^2-4ac\]

8–11 grade

Roots for D>0

NMT

\[x_1=\frac{-b-\sqrt D}{2a}\]

\[x_2=\frac{-b+\sqrt D}{2a}\]

Condition \(D\gt 0\)

8–11 grade

Root for D=0

NMT

\[x_1=x_2=-\frac b{2a}\]

Condition \(D=0\)

8–11 grade

Factorization of a quadratic trinomial

NMT

\[ax^2+bx+c=a(x-x_1)(x-x_2)\]

8–11 grade

Vieta's theorem

\[x_1+x_2=-\frac ba\]

\[x_1x_2=\frac ca\]

Condition \(a\ne0\)

8–11 grade

Inequality |f(x)|<a

\[|f(x)|\lt a\Longleftrightarrow -a\lt f(x)\lt a\]

Condition \(a\gt 0\)

8–11 grade

Inequality |f(x)|>a

\[|f(x)|\gt a\Longleftrightarrow f(x)\lt -a\ \text{або}\ f(x)\gt a\]

Condition \(a\ge0\)

8–11 grade

n-th term of an arithmetic progression

NMT

\[a_n=a_1+d(n-1)\]

9–11 grade

Sum of the first n terms of an arithmetic progression

NMT

\[S_n=\frac{a_1+a_n}{2}\,n\]

9–11 grade

Sum in terms of a₁ and d

\[S_n=\frac{2a_1+d(n-1)}2\,n\]

9–11 grade

Property of an arithmetic progression

\[a_n=\frac{a_{n-1}+a_{n+1}}2\]

9–11 grade

n-th term of a geometric progression

NMT

\[b_n=b_1q^{\,n-1}\]

9–11 grade

Sum of the first n terms of a geometric progression

NMT

\[S_n=\frac{b_1(q^n-1)}{q-1}\]

Condition \(q\ne1\)

9–11 grade

Sum of an infinite geometric progression

\[S=\frac{b_1}{1-q}\]

Condition \(|q|\lt 1\)

9–11 grade

Property of a geometric progression

\[b_n^2=b_{n-1}b_{n+1}\]

9–11 grade

x-coordinate of the parabola vertex

\[x_0=-\frac b{2a}\]

Condition \(a\ne0\)

8–11 grade

Vertex form of a quadratic function

\[y=a(x-x_0)^2+y_0\]

Condition \(a\ne0\)

8–11 grade

Linear function

\[y=kx+b\]

7–11 grade

Quadratic function

\[y=ax^2+bx+c\]

Condition \(a\ne0\)

8–11 grade

Power function

\[y=x^\alpha\]

9–11 grade

Exponential function

\[y=a^x\]

Condition \(a\gt 0,\ a\ne1\)

10–11 grade

Logarithmic function

\[y=\log_a x\]

Condition \(a\gt 0,\ a\ne1,\ x\gt 0\)

10–11 grade

Vertical graph translation

\[y=f(x)+b\]

8–11 grade

Horizontal graph translation

\[y=f(x-a)\]

8–11 grade

Stretch/compression along the x-axis

\[y=f(kx)\]

Condition \(k\ne0\)

8–11 grade

Stretch/compression along the y-axis

\[y=kf(x)\]

Condition \(k\ne0\)

8–11 grade

Reflection across the x-axis

\[y=-f(x)\]

8–11 grade

Reflection across the y-axis

\[y=f(-x)\]

8–11 grade

Definition of a logarithm

\[\log_a b=c\Longleftrightarrow a^c=b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0\)

10–11 grade

Logarithm of one

NMT

\[\log_a1=0\]

Condition \(a\gt 0,\ a\ne1\)

10–11 grade

Logarithm of the base

NMT

\[\log_a a=1\]

Condition \(a\gt 0,\ a\ne1\)

10–11 grade

Fundamental logarithmic identity

NMT

\[a^{\log_a b}=b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0\)

10–11 grade

Logarithm of a product

NMT

\[\log_a(bc)=\log_ab+\log_ac\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ c\gt 0\)

10–11 grade

Logarithm of a quotient

NMT

\[\log_a\frac bc=\log_ab-\log_ac\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ c\gt 0\)

10–11 grade

Logarithm of a power

NMT

\[\log_a b^n=n\log_a b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0\)

10–11 grade

Logarithm of a root

NMT

\[\log_a\sqrt[k]{b}=\frac1k\log_a b\]

Condition \(a\gt 0,\ a\ne1,\ b\gt 0,\ k\ne0\)

10–11 grade

Change of base

\[\log_a b=\frac{\log_c b}{\log_c a}\]

Condition \(a,b,c\gt 0,\ a\ne1,\ c\ne1\)

10–11 grade

Fundamental trigonometric identity

NMT

\[\sin^2\alpha+\cos^2\alpha=1\]

10–11 grade

Tangent

NMT

\[\tg\alpha=\frac{\sin\alpha}{\cos\alpha}\]

Condition \(\cos\alpha\ne0\)

10–11 grade

Cotangent

\[\ctg\alpha=\frac{\cos\alpha}{\sin\alpha}\]

Condition \(\sin\alpha\ne0\)

10–11 grade

Tangent identity

NMT

\[1+\tg^2\alpha=\frac1{\cos^2\alpha}\]

Condition \(\cos\alpha\ne0\)

10–11 grade

Cotangent identity

\[1+\ctg^2\alpha=\frac1{\sin^2\alpha}\]

Condition \(\sin\alpha\ne0\)

10–11 grade

Sine of a sum

\[\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\]

10–11 grade

Sine of a difference

\[\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\]

10–11 grade

Cosine of a sum

\[\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\]

10–11 grade

Cosine of a difference

\[\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\]

10–11 grade

Sine of a double angle

NMT

\[\sin2\alpha=2\sin\alpha\cos\alpha\]

10–11 grade

Cosine of a double angle

NMT

\[\cos2\alpha=\cos^2\alpha-\sin^2\alpha\]

\[\cos2\alpha=2\cos^2\alpha-1\]

\[\cos2\alpha=1-2\sin^2\alpha\]

10–11 grade

Tangent of a double angle

\[\tg2\alpha=\frac{2\tg\alpha}{1-\tg^2\alpha}\]

10–11 grade

Square of sine of a half-angle

\[\sin^2\frac\alpha2=\frac{1-\cos\alpha}{2}\]

10–11 grade

Square of cosine of a half-angle

\[\cos^2\frac\alpha2=\frac{1+\cos\alpha}{2}\]

10–11 grade

Formulas for 90°+α

NMT

\[\sin(90^\circ+\alpha)=\cos\alpha\]

\[\cos(90^\circ+\alpha)=-\sin\alpha\]

\[\tg(90^\circ+\alpha)=-\frac1{\tg\alpha}\]

10–11 grade

Formulas for 180°−α

NMT

\[\sin(180^\circ-\alpha)=\sin\alpha\]

\[\cos(180^\circ-\alpha)=-\cos\alpha\]

\[\tg(180^\circ-\alpha)=-\tg\alpha\]

10–11 grade

Equation sin x=a

\[x=(-1)^n\arcsin a+\pi n,\quad n\in\mathbb Z\]

Condition \(|a|\le1\)

10–11 grade

Equation cos x=a

\[x=\pm\arccos a+2\pi n,\quad n\in\mathbb Z\]

Condition \(|a|\le1\)

10–11 grade

Equation tg x=a

\[x=\arctg a+\pi n,\quad n\in\mathbb Z\]

10–11 grade

Derivative of a constant

NMT

\[(C)'=0\]

10–11 grade

Derivative of x

NMT

\[x'=1\]

10–11 grade

Derivative of a power function

NMT

\[(x^\alpha)'=\alpha x^{\alpha-1}\]

10–11 grade

Derivative of the square root

NMT

\[(\sqrt{x})'=\frac1{2\sqrt{x}}\]

10–11 grade

Derivative of e^x

NMT

\[(e^x)'=e^x\]

10–11 grade

Derivative of a^x

\[(a^x)'=a^x\ln a\]

10–11 grade

Derivative of ln x

NMT

\[(\ln x)'=\frac1x\]

10–11 grade

Derivative of log_a x

\[(\log_a x)'=\frac1{x\ln a}\]

10–11 grade

Derivative of sin x

NMT

\[(\sin x)'=\cos x\]

10–11 grade

Derivative of cos x

NMT

\[(\cos x)'=-\sin x\]

10–11 grade

Derivative of tg x

NMT

\[(\tg x)'=\frac1{\cos^2x}\]

10–11 grade

Derivative of ctg x

\[(\ctg x)'=-\frac1{\sin^2x}\]

10–11 grade

Derivative of a sum

NMT

\[(u+v)'=u'+v'\]

10–11 grade

Derivative of a difference

NMT

\[(u-v)'=u'-v'\]

10–11 grade

Derivative of a product

NMT

\[(uv)'=u'v+uv'\]

10–11 grade

Derivative of a quotient

NMT

\[\left(\frac uv\right)'=\frac{u'v-uv'}{v^2}\]

Condition \(v\ne0\)

10–11 grade

Constant factor rule

NMT

\[(Cu)'=Cu'\]

10–11 grade

Derivative of a composite function

\[(f(g(x)))'=f'(g(x))g'(x)\]

10–11 grade

Equation of a tangent line

\[y=f(x_0)+f'(x_0)(x-x_0)\]

10–11 grade

Antiderivative of 1

NMT

\[\int 1\,dx=x+C\]

11–11 grade

Antiderivative of a power function

NMT

\[\int x^\alpha dx=\frac{x^{\alpha+1}}{\alpha+1}+C,\quad \alpha\ne-1\]

11–11 grade

Antiderivative of 1/x

NMT

\[\int\frac{dx}{x}=\ln|x|+C\]

11–11 grade

Antiderivative of e^x

NMT

\[\int e^x dx=e^x+C\]

11–11 grade

Antiderivative of a^x

\[\int a^x dx=\frac{a^x}{\ln a}+C\]

11–11 grade

Antiderivative of sin x

NMT

\[\int\sin x\,dx=-\cos x+C\]

11–11 grade

Antiderivative of cos x

NMT

\[\int\cos x\,dx=\sin x+C\]

11–11 grade

Antiderivative of 1/cos²x

NMT

\[\int\frac{dx}{\cos^2x}=\tg x+C\]

11–11 grade

Antiderivative of 1/sin²x

\[\int\frac{dx}{\sin^2x}=-\ctg x+C\]

11–11 grade

Newton–Leibniz formula

NMT

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

11–11 grade

Area under a curve

\[S=\int_a^b f(x)\,dx\]

Condition \(f(x)\ge0\ \text{on}\ [a;b]\)

11–11 grade

Area between graphs

\[S=\int_a^b|f(x)-g(x)|\,dx\]

11–11 grade

Factorial

NMT

\[n!=1\cdot2\cdot\ldots\cdot n\]

11–11 grade

Permutations

NMT

\[P_n=n!\]

11–11 grade

k-permutations

NMT

\[A_n^k=\frac{n!}{(n-k)!}\]

11–11 grade

Combinations

NMT

\[C_n^k=\frac{n!}{k!(n-k)!}\]

11–11 grade

Classical probability

NMT

\[P(A)=\frac mn\]

m — number of favorable equally likely outcomes; n — total number of outcomes.

9–11 grade

Probability of the complementary event

\[P(\overline A)=1-P(A)\]

9–11 grade

Sum of probabilities of mutually exclusive events

\[P(A\cup B)=P(A)+P(B)\]

Condition \(A\cap B=\varnothing\)

10–11 grade

Product of probabilities of independent events

\[P(A\cap B)=P(A)P(B)\]

Condition \(A and B are independent\)

10–11 grade

Arithmetic mean

\[\bar x=\frac{x_1+x_2+\cdots+x_n}{n}\]

6–11 grade

Mean with frequencies

\[\bar x=\frac{x_1n_1+\cdots+x_kn_k}{n_1+\cdots+n_k}\]

9–11 grade

Range of a data set

\[R=x_{\max}-x_{\min}\]

9–11 grade