Awesome graphs that I made on desmos.com
Star:
Æ^ƨ0+1+0+0 2q∆72
≤Xp1+0+0+0 ∆36
p0+0+1+∞1 ∆36
Xq20+0+1+1 Ô.∆72
p0Ǻ1pº4É5q≤XÔ.X+0+1+∞1 Xpº2p2Ô.∆72
Ô.∆72
q∆36
p0+∞≤XÔ.X+1+∞1 XJ,
Heart:
*^*(1+0 Ǻ1q≤2Xq1
p¨Ô.2p0+1 Ç≤√X÷
Atom:
êa êé ÄÄPÄPÄÄÄ%Ä%ÄÄÄbÉSôi8ôW#ƒ16[#ƒ8[#ƒ4[#1#0:5#0:0584C
Smiley Face:
x^{2}+y^{2}\le1\left\{\left(x+0.4\right)^{2}+\left(y-0.4\right)^{2}>0.04,\left(x+0.4\right)^{2}+\left(y-0.4\right)^{2}<0.004\right\}\left\{\left(x-0.4\right)^{2}+\left(y-0.4\right)^{2}>0.04,\left(x-0.4\right)^{2}+\left(y-0.4\right)^{2}<0.004\right\}\left\{x^{2}>y+0.65,y>-0.35\right\}
Square:
\left|y\right|\le1\left\{\left|x\right|<1\right\}
Triangle:
y\ge0\left\{\left|x\right|<1-\frac{y}{\sqrt{3}}\right\}
Hexagon:
xy<1\left\{\left|y\right|<\sqrt{0.75}\right\}\left\{\frac{\left|y\right|}{2\sqrt{0.75}}<1-\left|x\right|\right\}
Octagon:
xy<1\left\{\left|x\right|<1\right\}\left\{\left|y\right|<1\right\}\left\{\left|x\right|+\left|y\right|<\sqrt{2}\right\}
Heart:
y>0\left\{\sin^{-1}\left(\left|x\right|\right)<y\right\}\left\{\sqrt{\left|x\right|-x^{2}}>y-\frac{\pi}{2}\right\}
Normal Distribution:
\operatorname{sech}x\ge y\left\{y>0\right\}
Area of any equilateral polygon:
\frac{na^{2}\tan\left(\frac{90n-180}{n}\right)}{4}
Hourglass:
\left|y\right|\ge\sin^{-1}\left(\left|x\right|\right)\left\{2ey^{2}+x^{2}<3\pi\right\}
Plus or minus symbol:
xy=xy^{2}\left\{\left|y-1\right|<0.75,\left|y\right|<1000x^{2}+0.0001\right\}\left\{\left|x\right|<0.75\right\}
Rounded Square:
x^{2}+y^{2}=ax^{2}y^{2}+1\left\{\left|x\right|\sqrt{a}<1.001\right\}\left\{\left|y\right|\sqrt{a}<1.001\right\}
Pixelated Circle
x^{2}+y^{2}\ge0\left\{2\left|\operatorname{round}\left(\left|ay\right|\right)-a\sqrt{1-\left(\frac{\operatorname{round}\left(ax\right)}{a}\right)^{2}}\right|\le1,2\left|\operatorname{round}\left(\left|ax\right|\right)-a\sqrt{1-\left(\frac{\operatorname{round}\left(ay\right)}{a}\right)^{2}}\right|\le1\right\}
Fleur-de-lis:
\left[\frac{\tan\left(\left|2x\right|\right)}{1.3}-0.67\left\{\left|x\right|<0.25\right\},5\left(\left|x\right|-0.45\right)^{2}-0.45\left\{0.75>\left|x\right|>0.25\right\},0.167-\sqrt{0.09-\left(\left|x\right|-0.5\right)^{2}}\left\{\left|x\right|<0.75\right\},\sqrt{0.16-\left(\left|x\right|-0.6\right)^{2}}+0.167,\sqrt{0.25-\left(\left|x\right|-0.75\right)^{2}}+0.595\left\{\left|x\right|>0.458\right\},0.595-\sqrt{0.25-\left(\left|x\right|-0.75\right)^{2}}\left\{\left|x\right|>1\right\},2-\frac{\tan\left(\left|2x\right|\right)}{1.3}\left\{\left|x\right|<\frac{\tan^{-1}1.3}{2}\right\}\right]
Checker pattern:
\cos x<\cos y
Rounded Square:
x^{2}+y^{2}=ax^{2}y^{2}+1+0\left(\sqrt{x\sqrt{a}+1.001}+\sqrt{1.001-x\sqrt{a}}+\sqrt{y\sqrt{a}+1.001}+\sqrt{1.001-y\sqrt{a}}+\sqrt{1-a}\right)
Square Wave:
\frac{\pi\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin\left(2\sin x\right)\right)\right)\right)\right)\right)\right)\right)\right)}{2}
Circle Grid:
\tan^{-1}\left(\cos y+\cos x\right)\ge1
Circle:
x^{2}+y^{2}\le1
Square:
\left|y\right|\le1+0x^{10^{10}}
Triangle:
y\ge0\left(\sqrt{\sqrt{3}\left(x+1\right)-y}\sqrt{\sqrt{3}\left(1-x\right)-y}\right)
Octagon:
xy<1+0\sqrt{1-x}\sqrt{x+1}\sqrt{1-y}\sqrt{y+1}\sqrt{x+y+\sqrt{2}}\sqrt{y-x+\sqrt{2}}\sqrt{\sqrt{2}-x-y}\sqrt{\sqrt{2}+x-y}
Heart:
y>0\sqrt{y-\sin^{-1}x}\sqrt{y-\sin^{-1}\left(-x\right)}\sqrt{\sqrt{\left|x\right|-x^{2}}-y+\frac{\pi}{2}}
Hexagon:
xy<1+0\sqrt{\sqrt{0.75}-y}\sqrt{y+\sqrt{0.75}}\sqrt{1-\frac{\left|y\right|}{2\sqrt{0.75}}+x}\sqrt{1-\frac{\left|y\right|}{2\sqrt{0.75}}-x}
Balloon:
2y-y^{2}\ge\sqrt{x^{2}+y^{2}}+x^{2}
Normal Distribution:
\operatorname{sech}x\ge y+\sqrt{y}
e:
\sum_{n=0}^{10!}\frac{1}{n!}
Phi (φ):
\frac{1+\sqrt{5}}{2}
Pi (π):
\cos^{-1}\left(-1\right)
Hyperbola:
x^{2}-y^{2}=1
Lemniscate:
x^{2}-y^{2}=\left(x^{2}+y^{2}\right)^{2}
Folium:
x^{3}+y^{3}=xy
Limacon:
\left(x^{2}+y^{2}-2x\right)^{2}=x^{2}+y^{2}
Rose:
\left(x^{2}+y^{2}\right)^{3}=\left(xy\right)^{2}
Parabola:
y=ax^{2}+bx+c
Vertex of a parabola:
\left(-\frac{b}{2a},c-\frac{b^{2}}{4a}\right)
Y intercept of a parabola:
\left(0,c\right)
X intercepts of a parabola:
\left(\frac{b+\sqrt{b^{2}-4ac}}{-2a},0\right),\left(\frac{\sqrt{b^{2}-4ac}-b}{2a},0\right)
Parabola from X intercepts:
x^{2}-x\left(x_{1}+x_{2}\right)+\frac{\left(x_{1}+x_{2}\right)^{2}-\left(x_{1}-x_{2}\right)^{2}}{4}
Parabola from 3 points:
y_{1}+x^{2}\frac{y_{3}-y_{2}-\frac{\left(x_{3}-x_{2}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}}{x_{3}^{2}-x_{2}^{2}-\frac{\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{3}-x_{2}\right)}{x_{2}-x_{1}}}+x\frac{y_{2}-y_{1}-\left(x_{2}^{2}-x_{1}^{2}\right)\frac{y_{3}-y_{2}-\frac{\left(x_{3}-x_{2}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}}{x_{3}^{2}-x_{2}^{2}-\frac{\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{3}-x_{2}\right)}{x_{2}-x_{1}}}}{x_{2}-x_{1}}-x_{1}^{2}\frac{y_{3}-y_{2}-\frac{\left(x_{3}-x_{2}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}}{x_{3}^{2}-x_{2}^{2}-\frac{\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{3}-x_{2}\right)}{x_{2}-x_{1}}}-x_{1}\frac{y_{2}-y_{1}-\left(x_{2}^{2}-x_{1}^{2}\right)\frac{y_{3}-y_{2}-\frac{\left(x_{3}-x_{2}\right)\left(y_{2}-y_{1}\right)}{x_{2}-x_{1}}}{x_{3}^{2}-x_{2}^{2}-\frac{\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{3}-x_{2}\right)}{x_{2}-x_{1}}}}{x_{2}-x_{1}}
Circle from 3 points:
\left(x+\frac{\left(x_{1}^{2}-x_{3}^{2}\right)\left(y_{1}-y_{2}\right)+\left(y_{1}^{2}-y_{3}^{2}\right)\left(y_{1}-y_{2}\right)+\left(x_{2}^{2}-x_{1}^{2}\right)\left(y_{1}-y_{3}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)\left(y_{1}-y_{3}\right)}{2\left(\left(x_{3}-x_{1}\right)\left(y_{1}-y_{2}\right)-\left(x_{2}-x_{1}\right)\left(y_{1}-y_{3}\right)\right)}\right)^{2}+\left(y+\frac{\left(x_{1}^{2}-x_{3}^{2}\right)\left(x_{1}-x_{2}\right)+\left(y_{1}^{2}-y_{3}^{2}\right)\left(x_{1}-x_{2}\right)+\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{1}-x_{3}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)\left(x_{1}-x_{3}\right)}{2\left(\left(y_{3}-y_{1}\right)\left(x_{1}-x_{2}\right)-\left(y_{2}-y_{1}\right)\left(x_{1}-x_{3}\right)\right)}\right)^{2}\le\left(x_{1}+\frac{\left(x_{1}^{2}-x_{3}^{2}\right)\left(y_{1}-y_{2}\right)+\left(y_{1}^{2}-y_{3}^{2}\right)\left(y_{1}-y_{2}\right)+\left(x_{2}^{2}-x_{1}^{2}\right)\left(y_{1}-y_{3}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)\left(y_{1}-y_{3}\right)}{2\left(\left(x_{3}-x_{1}\right)\left(y_{1}-y_{2}\right)-\left(x_{2}-x_{1}\right)\left(y_{1}-y_{3}\right)\right)}\right)^{2}+\left(y_{1}+\frac{\left(x_{1}^{2}-x_{3}^{2}\right)\left(x_{1}-x_{2}\right)+\left(y_{1}^{2}-y_{3}^{2}\right)\left(x_{1}-x_{2}\right)+\left(x_{2}^{2}-x_{1}^{2}\right)\left(x_{1}-x_{3}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)\left(x_{1}-x_{3}\right)}{2\left(\left(y_{3}-y_{1}\right)\left(x_{1}-x_{2}\right)-\left(y_{2}-y_{1}\right)\left(x_{1}-x_{3}\right)\right)}\right)^{2}
Hourglass:
\left|y\right|\ge\sin^{-1}\left(\left|x\right|\right)+0\sqrt{3\pi-2ey^{2}-x^{2}}
Interlocking ovals:
Face:
Interlocking Rectangles: