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📐 Algebra
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Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Slope Formula
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Point-Slope Form
$$y - y_1 = m(x - x_1)$$
Log Product Rule
$$\log_b(xy) = \log_b(x) + \log_b(y)$$
Log Quotient Rule
$$\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$$
Log Power Rule
$$\log_b(x^p) = p \cdot \log_b(x)$$
Difference of Squares
$$a^2 - b^2 = (a-b)(a+b)$$
Sum of Cubes
$$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$$
🔢 Sequences
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Arithmetic Nth Term
$$a_n = a_1 + (n-1)d$$
Arithmetic Sum
$$S_n = \frac{n}{2}(a_1 + a_n)$$
Geometric Nth Term
$$a_n = a_1 \cdot r^{n-1}$$
Finite Geometric Sum
$$S_n = \frac{a_1(1-r^n)}{1-r}$$
Infinite Geometric Sum (|r|<1)
$$S_\infty = \frac{a_1}{1-r}$$
🟩 Geometry
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Pythagorean Theorem
$$a^2 + b^2 = c^2$$
Distance Formula
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
Midpoint Formula
$$M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$$
Area of a Circle
$$A = \pi r^2$$
Circumference
$$C = 2\pi r$$
Volume of a Sphere
$$V = \frac{4}{3}\pi r^3$$
Volume of a Cylinder
$$V = \pi r^2 h$$
Volume of a Cone
$$V = \frac{1}{3}\pi r^2 h$$
Area of a Trapezoid
$$A = \frac{a+b}{2}h$$
⚠️ Trigonometry
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SOH CAH TOA
$$\sin = \frac{Op}{Hy}, \cos = \frac{Ad}{Hy}, \tan = \frac{Op}{Ad}$$
Common Angles
$$30^\circ=\frac{\pi}{6}, 45^\circ=\frac{\pi}{4}, 60^\circ=\frac{\pi}{3}$$
Pythagorean Identity
$$\sin^2\theta + \cos^2\theta = 1$$
Double Angle (Sin)
$$\sin(2\theta) = 2\sin\theta \cos\theta$$
Double Angle (Cos)
$$\cos(2\theta) = \cos^2\theta - \sin^2\theta$$
Law of Sines
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Law of Cosines
$$c^2 = a^2 + b^2 - 2ab\cos C$$
∫ Calculus
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Definition of Derivative
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
Power Rule
$$\frac{d}{dx}(x^n) = nx^{n-1}$$
Product Rule
$$(fg)' = f'g + fg'$$
Quotient Rule
$$(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$$
Chain Rule
$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$
Power Rule (Integral)
$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$
Integration by Parts
$$\int u dv = uv - \int v du$$
Fundamental Theorem
$$\int_a^b f(x)dx = F(b) - F(a)$$
📊 Statistics
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Mean (Average)
$$\bar{x} = \frac{\sum x_i}{n}$$
Standard Deviation (Sample)
$$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$
Z-Score
$$z = \frac{x - \mu}{\sigma}$$
Probability (Independent)
$$P(A \text{ and } B) = P(A) \cdot P(B)$$
Permutations
$$_nP_r = \frac{n!}{(n-r)!}$$
Combinations
$$_nC_r = \frac{n!}{r!(n-r)!}$$
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