vertices y=x^2+2x-3

{ "query": { "display": "vertices $$y=x^{2}+2x-3$$", "symbolab_question": "CONIC#vertices y=x^{2}+2x-3" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-1,-4)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=x^{2}+2x-3$$", "symbolab_question": "vertexquadratic y=x^{2}+2x-3" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=x^{2}+2x-3$$", "symbolab_question": "vertexparabola y=x^{2}+2x-3" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=x^{2}+2x-3$$", "symbolab_question": "vertexform y=x^{2}+2x-3" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=x^{2}+2x-3$$", "symbolab_question": "vertexzeros y=x^{2}+2x-3" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=x^{2}+2x-3:{\\quad}$$Minimum $$\\left(-1,\\:-4\\right)$$", "input": "y=x^{2}+2x-3", "steps": [ { "type": "definition", "title": "Parabola equation in polynomial form", "text": "The vertex of an up-down facing parabola of the form $$y=ax^2+bx+c\\:$$is $$x_{v}=-\\frac{b}{2a}$$" }, { "type": "step", "primary": "The parabola parameters are:", "result": "a=1,\\:b=2,\\:c=-3" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-1" }, { "type": "interim", "title": "Plug in $$x_{v}=-1\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=\\left(-1\\right)^{2}+2\\left(-1\\right)-3", "result": "y_{v}=-4", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-1\\right)^{2}+2\\left(-1\\right)-3:{\\quad}-4$$", "input": "\\left(-1\\right)^{2}+2\\left(-1\\right)-3", "result": "y_{v}=-4", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\left(-1\\right)^{2}-2\\cdot\\:1-3" }, { "type": "interim", "title": "$$\\left(-1\\right)^{2}=1$$", "input": "\\left(-1\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-1\\right)^{2}=1^{2}$$" ], "result": "=1^{2}" }, { "type": "step", "primary": "Apply rule $$1^{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s78E1FVQW6YvXK7raPRxih+c0ag8T1MwTer44+aCS/ZFAdx7pcd1x/bAWpIL8hAintf05A2GsVmPba4FjoW22b4iKyMg44e9p5G7GRfJ2en9g=" } }, { "type": "interim", "title": "$$2\\cdot\\:1=2$$", "input": "2\\cdot\\:1", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+PAYxXmLR4nDEBHt2jGqsN13jtrSFDx+UNsawjlOjV3jAewWnbvHwHHNJ9dhy4+WsiFkFLY2UCqxUg31WjmL6HgMxnm5w+rT3GGDPdan9jw=" } }, { "type": "step", "result": "=1-2-3" }, { "type": "step", "primary": "Subtract the numbers: $$1-2-3=-4$$", "result": "=-4" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zzuoR3/JnHaL2vl58S3gaDrfVeRXG9KjJU5mgumU/Bl5tMpJTBBccUWkSyvMe1SpIvsJv/VpmBmOllN+wMnDXXKF3u2OIb4bFA3EO8aRlSXM5rdwdO/6wbzT4FNnvlhePMuJclWpVka0IO+Hi/jfqQ==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-1,\\:-4\\right)" }, { "type": "step", "primary": "If $$a<0,\\:$$then the vertex is a maximum value<br/>If $$a>0,\\:$$then the vertex is a minimum value<br/>$$a=1$$", "result": "\\mathrm{Minimum}\\:\\left(-1,\\:-4\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-1))^{2}}{4\\frac{1}{4}}-4", "displayFormula": "4\\frac{1}{4}(y-(-4))=(x-(-1))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{17}{4}", "displayFormula": "y=-\\frac{17}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-1,-4)", "(-1,-\\frac{15}{4})" ], "pointsDecimal": [ { "fst": -1, "snd": -4 }, { "fst": -1, "snd": -3.75 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": 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