{"version":9,"randomSeed":"7591b3b9e61d395a6d48304d5f3c44cb","graph":{"viewport":{"xmin":-0.1246171375225915,"ymin":-0.07077860348358667,"xmax":2.7494966399840473,"ymax":1.712386352842713}},"expressions":{"list":[{"type":"folder","id":"80","title":"commentary","collapsed":true},{"type":"text","id":"85","folderId":"80","text":"theoretically, I could derive coefficients [a, b, c] from the points with a direct formula"},{"type":"text","id":"87","folderId":"80","text":"as in https://www.desmos.com/calculator/snimp9lxvu"},{"type":"text","id":"89","folderId":"80","text":"(one would have to modify that one a bit)"},{"type":"text","id":"91","folderId":"80","text":"but the point here is finding extrema, not quadratic algebra"},{"type":"text","id":"93","folderId":"80","text":"so I'll just use Desmos regression"},{"type":"folder","id":"71","title":"functions","collapsed":true},{"type":"text","id":"95","folderId":"71","text":"function of which to find extrema"},{"type":"expression","id":"1","folderId":"71","color":"#c74440","latex":"f\\left(x\\right)=\\sin x"},{"type":"text","id":"119","folderId":"71","text":"see https://www.desmos.com/calculator/clfyruuewl"},{"type":"text","id":"116","folderId":"71","text":"determinant (row-major)"},{"type":"expression","id":"117","folderId":"71","color":"#c74440","latex":"D\\left(M\\right)=M\\left[1\\right]M\\left[5\\right]M\\left[9\\right]+M\\left[2\\right]M\\left[6\\right]M\\left[7\\right]+M\\left[3\\right]M\\left[4\\right]M\\left[8\\right]-M\\left[3\\right]M\\left[5\\right]M\\left[7\\right]-M\\left[2\\right]M\\left[4\\right]M\\left[9\\right]-M\\left[1\\right]M\\left[6\\right]M\\left[8\\right]"},{"type":"text","id":"121","folderId":"71","text":"coefficients for parabola (three-element list) by points"},{"type":"expression","id":"128","folderId":"71","color":"#000000","latex":"C\\left(a,b,c\\right)=\\frac{\\left[D\\left(\\left[a.y,a.x,1,b.y,b.x,1,c.y,c.x,1\\right]\\right),D\\left(\\left[a.x^{2},a.y,1,b.x^{2},b.y,1,c.x^{2},c.y,1\\right]\\right),D\\left(\\left[a.x^{2},a.x,a.y,b.x^{2},b.x,b.y,c.x^{2},c.x,c.y\\right]\\right)\\right]}{D\\left(\\left[a.x^{2},a.x,1,b.x^{2},b.x,1,c.x^{2},c.x,1\\right]\\right)}"},{"type":"text","id":"97","folderId":"71","text":"vertex (x, y) from parabola coefficients"},{"type":"expression","id":"98","folderId":"71","color":"#c74440","latex":"V\\left(a,b,c\\right)=\\left(-\\frac{b}{2a},-\\frac{b^{2}}{4a}+c\\right)"},{"type":"text","id":"104","folderId":"71","text":"three elements in increasing order from L (TEIIO) and p"},{"type":"expression","id":"102","folderId":"71","color":"#6042a6","latex":"S\\left(L,p\\right)=\\left[L\\left[2\\right],p,L\\left[3\\right]\\right]"},{"type":"folder","id":"77","title":"step 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