Pre-Calculus

Cheatsheet Content

### Functions - **Definition:** A relation where each input $x$ has exactly one output $y$. - Notation: $y = f(x)$ - **Domain:** Set of all possible input values ($x$). - **Range:** Set of all possible output values ($y$). - **Vertical Line Test:** Used to determine if a graph represents a function. - **Types of Functions:** - **Linear:** $f(x) = mx + b$ - **Quadratic:** $f(x) = ax^2 + bx + c$ - **Polynomial:** $f(x) = a_n x^n + ... + a_1 x + a_0$ - **Rational:** $f(x) = \frac{P(x)}{Q(x)}$, $Q(x) \neq 0$ - **Exponential:** $f(x) = a^x$, $a > 0, a \neq 1$ - **Logarithmic:** $f(x) = \log_a x$, $a > 0, a \neq 1$ (inverse of exponential) - **Function Operations:** - **Sum:** $(f+g)(x) = f(x) + g(x)$ - **Difference:** $(f-g)(x) = f(x) - g(x)$ - **Product:** $(fg)(x) = f(x)g(x)$ - **Quotient:** $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$ - **Composition:** $(f \circ g)(x) = f(g(x))$ #### Inverse Functions - **Definition:** If $f(g(x)) = x$ and $g(f(x)) = x$, then $f$ and $g$ are inverse functions. Denoted $f^{-1}(x)$. - **Horizontal Line Test:** Used to determine if a function has an inverse. - **Finding Inverse:** 1. Replace $f(x)$ with $y$. 2. Swap $x$ and $y$. 3. Solve for $y$. 4. Replace $y$ with $f^{-1}(x)$. ### Polynomial & Rational Functions #### Polynomial Functions - **Degree:** Highest power of $x$. - **Leading Coefficient:** Coefficient of the term with the highest power. - **End Behavior:** Determined by degree and leading coefficient. - **Zeros/Roots:** Values of $x$ for which $f(x) = 0$. - **Factor Theorem:** $(x-c)$ is a factor if $f(c) = 0$. - **Remainder Theorem:** If a polynomial $f(x)$ is divided by $(x-c)$, the remainder is $f(c)$. - **Rational Root Theorem:** Possible rational roots are $\frac{p}{q}$, where $p$ divides constant term and $q$ divides leading coefficient. - **Multiplicity of Zeros:** - Even multiplicity: graph touches x-axis and turns around. - Odd multiplicity: graph crosses x-axis. #### Rational Functions - **Asymptotes:** - **Vertical Asymptote (VA):** Occurs where denominator is zero and numerator is non-zero (or has a lower power). - **Horizontal Asymptote (HA):** - Degree of numerator Degree of denominator: No HA (Slant/Oblique Asymptote). - **Slant/Oblique Asymptote:** Occurs when degree of numerator is exactly one more than degree of denominator. Found by polynomial long division. - **Holes:** Occur when a factor cancels out from numerator and denominator. ### Exponential & Logarithmic Functions #### Exponential Functions - **Form:** $f(x) = a^x$, where $a > 0, a \neq 1$. - **Properties:** - If $a > 1$, increasing. - If $0 0, a \neq 1$. - **Relation to Exponential:** $y = \log_a x \iff a^y = x$. - **Properties:** - Domain: $(0, \infty)$, Range: $(-\infty, \infty)$. - Vertical asymptote at $x=0$. - $\log_a 1 = 0$ - $\log_a a = 1$ - $\log_a (MN) = \log_a M + \log_a N$ - $\log_a (\frac{M}{N}) = \log_a M - \log_a N$ - $\log_a (M^p) = p \log_a M$ - **Change of Base Formula:** $\log_a x = \frac{\log_b x}{\log_b a} = \frac{\ln x}{\ln a} = \frac{\log x}{\log a}$. - **Natural Logarithm:** $\ln x = \log_e x$. - **Common Logarithm:** $\log x = \log_{10} x$. ### Trigonometry Basics #### Right Triangle Trigonometry - **SOH CAH TOA:** - $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ - **Reciprocal Identities:** - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\cot \theta = \frac{1}{\tan \theta}$ #### Unit Circle - Circle with radius 1 centered at origin. - $(x,y) = (\cos \theta, \sin \theta)$ - $\tan \theta = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}$ #### Special Angles | Angle ($\theta$) | Radians | $\sin \theta$ | $\cos \theta$ | $\tan \theta$ | |------------------|---------|---------------|---------------|---------------| | $0^\circ$ | $0$ | $0$ | $1$ | $0$ | | $30^\circ$ | $\pi/6$ | $1/2$ | $\sqrt{3}/2$ | $1/\sqrt{3}$ | | $45^\circ$ | $\pi/4$ | $\sqrt{2}/2$ | $\sqrt{2}/2$ | $1$ | | $60^\circ$ | $\pi/3$ | $\sqrt{3}/2$ | $1/2$ | $\sqrt{3}$ | | $90^\circ$ | $\pi/2$ | $1$ | $0$ | Undefined | #### Radian-Degree Conversion - $180^\circ = \pi$ radians - Degrees to Radians: Multiply by $\frac{\pi}{180^\circ}$ - Radians to Degrees: Multiply by $\frac{180^\circ}{\pi}$ #### Fundamental Identities - **Pythagorean Identities:** - $\sin^2 \theta + \cos^2 \theta = 1$ - $1 + \tan^2 \theta = \sec^2 \theta$ - $1 + \cot^2 \theta = \csc^2 \theta$ - **Quotient Identity:** $\tan \theta = \frac{\sin \theta}{\cos \theta}$ ### Trigonometric Graphs #### General Forms - $y = A \sin(Bx - C) + D$ - $y = A \cos(Bx - C) + D$ - **Amplitude:** $|A|$ (vertical stretch) - **Period:** $\frac{2\pi}{|B|}$ (horizontal stretch/compression) - **Phase Shift:** $\frac{C}{B}$ (horizontal shift: right if $C>0$, left if $C ### Trigonometric Identities & Formulas #### Sum and Difference Formulas - $\sin(u \pm v) = \sin u \cos v \pm \cos u \sin v$ - $\cos(u \pm v) = \cos u \cos v \mp \sin u \sin v$ - $\tan(u \pm v) = \frac{\tan u \pm \tan v}{1 \mp \tan u \tan v}$ #### Double Angle Formulas - $\sin(2u) = 2 \sin u \cos u$ - $\cos(2u) = \cos^2 u - \sin^2 u = 2\cos^2 u - 1 = 1 - 2\sin^2 u$ - $\tan(2u) = \frac{2\tan u}{1 - \tan^2 u}$ #### Half Angle Formulas - $\sin(\frac{u}{2}) = \pm \sqrt{\frac{1 - \cos u}{2}}$ - $\cos(\frac{u}{2}) = \pm \sqrt{\frac{1 + \cos u}{2}}$ - $\tan(\frac{u}{2}) = \frac{1 - \cos u}{\sin u} = \frac{\sin u}{1 + \cos u}$ #### Law of Sines & Cosines - **Law of Sines:** $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ - **Law of Cosines:** - $a^2 = b^2 + c^2 - 2bc \cos A$ - $b^2 = a^2 + c^2 - 2ac \cos B$ - $c^2 = a^2 + b^2 - 2ab \cos C$ ### Sequences & Series #### Sequences - **Arithmetic Sequence:** Each term differs by a common difference $d$. - $a_n = a_1 + (n-1)d$ - **Geometric Sequence:** Each term is multiplied by a common ratio $r$. - $a_n = a_1 r^{n-1}$ #### Series - **Arithmetic Series (Sum):** $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ - **Geometric Series (Sum):** $S_n = \frac{a_1(1 - r^n)}{1 - r}$, $r \neq 1$ - **Infinite Geometric Series (Sum):** $S = \frac{a_1}{1 - r}$, if $|r| ### Conic Sections #### General Form - $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ #### Circle - $(x-h)^2 + (y-k)^2 = r^2$ - Center: $(h,k)$, Radius: $r$ #### Parabola - **Vertical:** $(x-h)^2 = 4p(y-k)$ - Vertex: $(h,k)$, Focus: $(h, k+p)$, Directrix: $y = k-p$ - **Horizontal:** $(y-k)^2 = 4p(x-h)$ - Vertex: $(h,k)$, Focus: $(h+p, k)$, Directrix: $x = h-p$ #### Ellipse - **Horizontal Major Axis:** $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, $a > b$ - **Vertical Major Axis:** $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, $a > b$ - Center: $(h,k)$ - Vertices: $\pm a$ from center along major axis. - Covertices: $\pm b$ from center along minor axis. - Foci: $\pm c$ from center along major axis, where $c^2 = a^2 - b^2$. #### Hyperbola - **Horizontal Transverse Axis:** $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ - **Vertical Transverse Axis:** $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ - Center: $(h,k)$ - Vertices: $\pm a$ from center along transverse axis. - Foci: $\pm c$ from center along transverse axis, where $c^2 = a^2 + b^2$. - Asymptotes (for horizontal): $y-k = \pm \frac{b}{a}(x-h)$ - Asymptotes (for vertical): $y-k = \pm \frac{a}{b}(x-h)$ ### Matrices & Determinants #### Basic Operations - **Addition/Subtraction:** Add/subtract corresponding elements. Matrices must be same dimensions. - **Scalar Multiplication:** Multiply every element by the scalar. - **Matrix Multiplication:** If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$. - $(AB)_{ij} = \sum_{k=1}^n A_{ik} B_{kj}$ - Not commutative ($AB \neq BA$ in general). #### Determinants - **$2 \times 2$ Matrix:** If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $\det(A) = ad - bc$. - **$3 \times 3$ Matrix:** Use cofactor expansion. - **Properties:** - $\det(I) = 1$ (Identity matrix) - $\det(A^T) = \det(A)$ - $\det(AB) = \det(A)\det(B)$ - If $\det(A) = 0$, matrix is singular (no inverse). #### Inverse Matrix - **$2 \times 2$ Inverse:** If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad-bc \neq 0$. - **Solving Systems using Inverse:** $AX = B \implies X = A^{-1}B$. ### Vectors #### Basic Concepts - **Definition:** Quantity with both magnitude and direction. - **Component Form:** $\vec{v} = \langle v_1, v_2 \rangle$ (2D) or $\langle v_1, v_2, v_3 \rangle$ (3D). - **Magnitude:** $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$ (2D). - **Unit Vector:** $\hat{u} = \frac{\vec{v}}{|\vec{v}|}$ (vector with magnitude 1 in the same direction). #### Operations - **Addition:** $\langle v_1, v_2 \rangle + \langle u_1, u_2 \rangle = \langle v_1+u_1, v_2+u_2 \rangle$. - **Scalar Multiplication:** $c\langle v_1, v_2 \rangle = \langle cv_1, cv_2 \rangle$. - **Dot Product:** $\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 = |\vec{u}||\vec{v}|\cos\theta$. - If $\vec{u} \cdot \vec{v} = 0$, vectors are orthogonal. - **Cross Product (3D only):** $\vec{u} \times \vec{v}$ results in a vector orthogonal to both $\vec{u}$ and $\vec{v}$. - Magnitude: $|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin\theta$. ### Polar Coordinates & Complex Numbers #### Polar Coordinates - **$(r, \theta)$:** $r$ is distance from origin, $\theta$ is angle from positive x-axis. - **Conversion:** - **Polar to Rectangular:** $x = r \cos \theta$, $y = r \sin \theta$. - **Rectangular to Polar:** $r = \sqrt{x^2 + y^2}$, $\tan \theta = \frac{y}{x}$ (mind the quadrant for $\theta$). #### Complex Numbers - **Form:** $z = a + bi$, where $i = \sqrt{-1}$. - **Complex Plane:** Real part ($a$) on x-axis, Imaginary part ($b$) on y-axis. - **Operations:** - **Addition/Subtraction:** $(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i$. - **Multiplication:** $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$. - **Division:** $\frac{a+bi}{c+di} = \frac{a+bi}{c+di} \cdot \frac{c-di}{c-di}$. - **Complex Conjugate:** $\bar{z} = a - bi$. - **Absolute Value/Modulus:** $|z| = \sqrt{a^2 + b^2}$. - **Polar Form:** $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$ and $\tan \theta = \frac{b}{a}$. - **De Moivre's Theorem:** $[r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta))$.