Formula sheets.

Elementary math

Tutoring

Add fractions

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$

Multiply fractions

$$\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$$

Divide fractions

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$$

Area of a rectangle

$$A=\ell w$$

Perimeter of a rectangle

$$P=2\ell +2w$$

Area of a triangle

$$A=\frac{1}{2}bh$$

Middle school math

Tutoring

Percent

$$\text{part}=\text{percent}\times \text{whole}$$

Percent change

$$\frac{\text{new}-\text{old}}{\text{old}}$$

Unit rate

$$\text{rate}=\frac{\text{amount}}{\text{units}}$$

Proportion

$$\frac{a}{b}=\frac{c}{d}⟺ad=bc$$

Mean

$$\text{mean}=\frac{\text{sum}}{\text{count}}$$

Distance

$$d=rt$$

Algebra

Tutoring

Slope

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Slope-intercept form

$$y=mx+b$$

Point-slope form

$$y-y_1=m(x-x_1)$$

Quadratic formula

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Difference of squares

$$a^2-b^2=(a-b)(a+b)$$

Perfect square

$$(a+b{)}^2=a^2+2ab+b^2$$

Product of powers

$$x^a{x}^b=x^{a+b}$$

Power of a power

$$(x^a{)}^b=x^{ab}$$

Geometry and trigonometry

Tutoring

Pythagorean theorem

$$a^2+b^2=c^2$$

Circle circumference

$$C=2\pi r$$

Circle area

$$A=\pi r^2$$

Sector area

$$A=\frac{\theta }{360^{\circ }}\pi r^2$$

Distance between points

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Midpoint

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

Cylinder volume

$$V=\pi r^{2}h$$

Sphere volume

$$V=\frac{4}{3}\pi r^3$$

Law of Sines

$$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$

Law of Cosines

$$c^2=a^2+b^2-2ab\cos C$$

Precalculus

Tutoring

Log definition

$${\log }_{b}y=x⟺b^x=y$$

Log of a product

$$\log (ab)=\log a+\log b$$

Log of a quotient

$$\log \frac{a}{b}=\log a-\log b$$

Log of a power

$$\log a^n=n\log a$$

Change of base

$${\log }_{b}a=\frac{\ln a}{\ln b}$$

Pythagorean identity

$${\sin }^2\theta +{\cos }^2\theta =1$$

Double angle

$$\sin 2\theta =2\sin \theta \cos \theta$$

Compound interest

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}$$

AP and college calculus

Tutoring

Power rule

$$\frac{d}{dx}{x}^n=n{x}^{n-1}$$

Product rule

$$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$$

Quotient rule

$${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$$

Chain rule

$$\frac{d}{dx}f(g(x))=f^{\prime }(g(x))g^{\prime }(x)$$

Trig derivatives

$$(\sin x{)}^{\prime }=\cos x,(\cos x{)}^{\prime }=-\sin x$$

Exponential and log

$$(e^x{)}^{\prime }=e^x,(\ln x{)}^{\prime }=\frac{1}{x}$$

Power rule for integrals

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1$$

Fundamental theorem

$${\int }_a^{b}f(x)dx=F(b)-F(a)$$

SAT Math

Tutoring

Linear model

$$y=mx+b$$

Discriminant

$$b^2-4ac$$

Vertex x-value

$$x=-\frac{b}{2a}$$

Exponential growth

$$y=a(1+r{)}^t$$

Percent decrease

$$\text{new}=\text{old}(1-r)$$

Circle equation

$$(x-h{)}^2+(y-k{)}^2=r^2$$

ACT Math

Tutoring

SOHCAHTOA: sine

$$\sin \theta =\frac{\text{opp}}{\text{hyp}}$$

SOHCAHTOA: cosine

$$\cos \theta =\frac{\text{adj}}{\text{hyp}}$$

SOHCAHTOA: tangent

$$\tan \theta =\frac{\text{opp}}{\text{adj}}$$

Probability

$$P=\frac{\text{wanted}}{\text{total}}$$

Arithmetic sequence

$$a_n=a_1+(n-1)d$$

Geometric sequence

$$a_n=a_1{r}^{n-1}$$

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