Pre-Calculus
Master Quiz

\[ f \text{ is a function} \iff \forall x, \text{ exactly one } y \] \[ \text{Vertical Line Test: passes} \Rightarrow \text{function} \] \[ (f \circ g)(x) = f(g(x)) \] \[ f^{-1} \text{ exists} \iff f \text{ is one-to-one (passes HLT)} \]

\[\text{Remainder Theorem: } f(a) = \text{remainder when } f(x) \div (x-a)\] \[\text{Factor Theorem: } (x-a) \text{ is a factor} \iff f(a)=0\] \[\text{Rational Root Test: possible roots} = \pm\frac{p}{q}\]

\[\log_b(MN)=\log_b M+\log_b N\] \[\log_b\!\left(\tfrac{M}{N}\right)=\log_b M-\log_b N\] \[\log_b(M^p)=p\log_b M\] \[\log_b M=\frac{\ln M}{\ln b}\quad\text{(Change of Base)}\] \[A=Pe^{rt}\quad\text{(Continuous Compound)}\]

\[\sin^2\theta+\cos^2\theta=1\quad\tan\theta=\frac{\sin\theta}{\cos\theta}\] \[\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\] \[\cos(2\theta)=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1\] \[\text{Law of Sines: }\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\] \[\text{Law of Cosines: }c^2=a^2+b^2-2ab\cos C\]

\[\text{Circle: }(x-h)^2+(y-k)^2=r^2\] \[\text{Parabola: }(x-h)^2=4p(y-k)\text{ (vertical)}\] \[\text{Ellipse: }\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1,\quad a>b>0,\quad c^2=a^2-b^2\] \[\text{Hyperbola: }\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,\quad c^2=a^2+b^2\]

\[a_n=a_1+(n-1)d\quad\text{(Arithmetic)}\] \[S_n=\frac{n}{2}(a_1+a_n)\quad\text{(Arithmetic Sum)}\] \[a_n=a_1\cdot r^{n-1}\quad\text{(Geometric)}\] \[S_n=a_1\cdot\frac{1-r^n}{1-r}\quad(r\neq1)\quad\text{(Geometric Sum)}\] \[S_\infty=\frac{a_1}{1-r}\quad(|r|<1)\quad\text{(Infinite Geometric)}\]

\[\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2=|\vec{u}||\vec{v}|\cos\theta\] \[\text{det}\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc\] \[A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]

\[\lim_{x\to a}f(x)=L\iff\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\] \[\lim_{x\to\infty}\frac{p(x)}{q(x)}: \text{ compare degrees of numerator \& denominator}\]