1 functions and graphs

1.1 modulus function

\[|f(x)|=\begin{cases} -f(x) & ,f(x)\lt{}0 \\ f(x) & ,f(x)\geq{}0 \end{cases}\]

it can also be represented as \(|x|=\sqrt{x^2}\)

1.2 conic sections

1.2.1 circles

\[(x-h)^2+(y-k)^2=r^2\]

circle with center \((h,k)\) and radius \(r\)

1.2.2 ellipses

\[(\frac{x-h}{a})^2+(\frac{y-k}{b})^2=1\]

ellipse with center \((h,k)\), width \(2a\) and height \(2b\)

1.2.3 hyperbolas

\[(\frac{x-h}{a})^2-(\frac{y-k}{b})^2=1\] \[-(\frac{x-h}{a})^2+(\frac{y-k}{b})^2=1\]

asymptotes: \(y=\pm\frac{b}{a}(x-h)+k\)

1.3 inverse circular functions

function	domain	range
\(\arcsin(x)\)	\([0,1]\)	\([-\frac{\pi}{2},\frac{\pi}{2}]\)
\(\arccos(x)\)	\([0,1]\)	\([0,\pi]\)
\(\arctan(x)\)	\([-\infty,\infty]\)	\([-\frac{\pi}{2},\frac{\pi}{2}]\)

1.4 parametric equations

\[x=f(t)\] \[y=g(t)\]

use simultaneous equation to solve parametric equations