{"version":9,"randomSeed":"7dde9c4a1b05f943f8c6eeb2354bbe61","graph":{"viewport":{"xmin":-4.2203015484372415,"ymin":-10.489004821489646,"xmax":21.268284877344016,"ymax":11.130547897957078}},"expressions":{"list":[{"type":"folder","id":"2","title":"original points"},{"type":"expression","id":"8","folderId":"2","color":"#2d70b3","latex":"A=\\left(3.9,1.1\\right)"},{"type":"expression","id":"9","folderId":"2","color":"#388c46","latex":"B=\\left(11.55,2.7\\right)"},{"type":"expression","id":"10","folderId":"2","color":"#6042a6","latex":"C=\\left(9.43,-0.88\\right)"},{"type":"folder","id":"12","title":"quadratic function"},{"type":"text","id":"17","folderId":"12","text":"see https://www.desmos.com/calculator/snimp9lxvu"},{"type":"expression","id":"13","folderId":"12","color":"#c74440","latex":"P=B-A","hidden":true},{"type":"expression","id":"14","folderId":"12","color":"#2d70b3","latex":"Q=C-A","hidden":true},{"type":"expression","id":"15","folderId":"12","color":"#388c46","latex":"k=\\frac{P.x\\cdot Q.y-P.y\\cdot Q.x}{P.x\\cdot Q.x\\cdot\\left(Q.x-P.x\\right)}"},{"type":"expression","id":"18","folderId":"12","color":"#000000","latex":"m=\\frac{Q.y-k\\left(Q.x^{2}\\right)}{Q.x}"},{"type":"expression","id":"19","folderId":"12","color":"#c74440","latex":"k\\left(x-A.x\\right)^{2}+m\\left(x-A.x\\right)+A.y","hidden":true},{"type":"expression","id":"23","folderId":"12","color":"#6042a6","latex":"kx^{2}-\\left(2kA.x\\right)x+mx-mA.x+A.y+kA.x^{2}","hidden":true},{"type":"expression","id":"24","folderId":"12","color":"#000000","latex":"a=k"},{"type":"expression","id":"25","folderId":"12","color":"#c74440","latex":"b=-2kA.x+m"},{"type":"expression","id":"26","folderId":"12","color":"#2d70b3","latex":"c=kA.x^{2}-mA.x+A.y"},{"type":"expression","id":"27","folderId":"12","color":"#388c46","latex":"f\\left(x\\right)=ax^{2}+bx+c"},{"type":"folder","id":"21","title":"roots of function"},{"type":"expression","id":"22","folderId":"21","color":"#388c46","latex":"r_{1}=\\frac{-b+\\sqrt{b^{2}-4ac}}{2a}"},{"type":"expression","id":"28","folderId":"21","color":"#6042a6","latex":"r_{2}=\\frac{-b-\\sqrt{b^{2}-4ac}}{2a}"},{"type":"expression","id":"30","folderId":"21","color":"#c74440","latex":"\\left(\\left[r_{1},r_{2}\\right],0\\right)"}]}}