(y-(-1))^2=2(x-1)

{ "query": { "display": "$$\\left(y-\\left(-1\\right)\\right)^{2}=2\\left(x-1\\right)$$", "symbolab_question": "CONIC#(y-(-1))^{2}=2(x-1)" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(1,-1),p=\\frac{1}{2}" }, "steps": { "type": "interim", "title": "$$\\left(y-\\left(-1\\right)\\right)^{2}=2\\left(x-1\\right):\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(1,\\:-1\\right),\\:$$and focal length $$|p|=\\frac{1}{2}$$", "input": "\\left(y-\\left(-1\\right)\\right)^{2}=2\\left(x-1\\right)", "steps": [ { "type": "definition", "title": "Parabola standard equation", "text": "$$4p\\left(x-h\\right)=\\left(y-k\\right)^{2}\\:$$ is the standard equation for a right-left facing parabola with vertex at $$\\left(h,\\:k\\right),\\:$$<br/>and a focal length $$|p|$$" }, { "type": "interim", "title": "Rewrite $$\\left(y-\\left(-1\\right)\\right)^{2}=2\\left(x-1\\right)\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{2}\\left(x-1\\right)=\\left(y-\\left(-1\\right)\\right)^{2}$$", "input": "\\left(y-\\left(-1\\right)\\right)^{2}=2\\left(x-1\\right)", "steps": [ { "type": "step", "primary": "Switch sides", "result": "2\\left(x-1\\right)=\\left(y-\\left(-1\\right)\\right)^{2}" }, { "type": "step", "primary": "Factor $$2$$", "result": "2\\left(x+\\frac{-2}{2}\\right)=\\left(y-\\left(-1\\right)\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "2\\left(x-1\\right)=\\left(y-\\left(-1\\right)\\right)^{2}" }, { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{2}{4}\\left(x-1\\right)=\\left(y-\\left(-1\\right)\\right)^{2}" }, { "type": "step", "primary": "Simplify", "result": "4\\cdot\\:\\frac{1}{2}\\left(x-1\\right)=\\left(y-\\left(-1\\right)\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7x7c70iKgri0FohdBA3wVDTQw7MNONbtupZ5amwm1UvUmnIPvAXZpEx2L4ux3cQBbx76H4czezJl99cnQ2vKI4SNyQCASdkcKydARUOaLOvsYmi/QRr/a+1JySSnsiH5LdcVxc9HCPCyzZlJGlG1xKaYN6z/6D5StsnFwXE8BrNwRztKE552b3ssyeALQtWRjdllRtXcbobcHL3OfgLNk6O8r5YU2ZsAeB241LtFBNTNXa3MeiSeRf1gRQdFDf9qEe77HpL4nTQimDoQK8+yrQlM61C7GItjVtv6yXm2ATwU=" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(1,\\:-1\\right),\\:p=\\frac{1}{2}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{2}(x-1)}-1", "displayFormula": "4\\frac{1}{2}(x-1)=(y-(-1))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{2}(x-1)}-1", "displayFormula": "4\\frac{1}{2}(x-1)=(y-(-1))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=\\frac{1}{2}", "displayFormula": "x=\\frac{1}{2}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(1,-1)", "(\\frac{3}{2},-1)" ], "pointsDecimal": [ { "fst": 1, "snd": -1 }, { "fst": 1.5, "snd": -1 } ], "attributes": [ { "color": "PURPLE", "labels": [ "\\mathrm{vertex}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] }, { "color": "PURPLE", "labels": [ "\\mathrm{focus}" ], "labelTypes": [ "DEFAULT" ], "labelColors": [ "PURPLE" ] } ] }, "functionChanges": [ { "origFormulaLatex": [], "finalFormulaLatex": [], "plotTitle": "4\\cdot \\frac{1}{2}(x-1)=(y-(-1))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -4.625, "xMax": 6.625, "yMin": -6.625, "yMax": 4.625 } }, "showViewLarger": true } } }