x+3=y^2

{ "query": { "display": "$$x+3=y^{2}$$", "symbolab_question": "CONIC#x+3=y^{2}" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "formula", "default": "(h,k)=(-3,0),p=\\frac{1}{4}" }, "steps": { "type": "interim", "title": "$$x+3=y^{2}:\\quad$$Parabola with vertex at $$\\left(h,\\:k\\right)=\\left(-3,\\:0\\right),\\:$$and focal length $$|p|=\\frac{1}{4}$$", "input": "x+3=y^{2}", "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 $$x+3=y^{2}\\:$$in the standard form:$${\\quad}4\\cdot\\:\\frac{1}{4}\\left(x-\\left(-3\\right)\\right)=\\left(y-0\\right)^{2}$$", "input": "x+3=y^{2}", "steps": [ { "type": "step", "primary": "Factor $$4$$", "result": "4\\cdot\\:\\frac{1}{4}\\left(x+3\\right)=y^{2}" }, { "type": "step", "primary": "Rewrite as", "result": "4\\cdot\\:\\frac{1}{4}\\left(x-\\left(-3\\right)\\right)=\\left(y-0\\right)^{2}" } ], "meta": { "interimType": "Parabola Canonical Format Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7XyLbZXp+Wbh+m4tF03n6J4xWrU5J8tK8xKilMipHyFleTXjeP8Fry4AS6VRPW/JxnSd9ckJvhaVSw11yD/KmDHfDBptXBowSMDY4Q9A3bJAIFkbUz7puWZSuakBtmZXzbFhjH/Na3u3ce/JmSdVjdixts5630ugtF+sLZdTgt/F6pfF1z6umzUJTJvt+ojYZ/qbA2SHa7igUO0C4cX6igGTjTKQwxlwZHqyMj3rDI4JpGwoPd870QszKiiHupZ++ihjOSo1xKKuRxYOAfpknTA==" } }, { "type": "step", "primary": "Therefore parabola properties are:", "result": "\\left(h,\\:k\\right)=\\left(-3,\\:0\\right),\\:p=\\frac{1}{4}" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\sqrt{4\\frac{1}{4}(x-(-3))}+0", "displayFormula": "4\\frac{1}{4}(x-(-3))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\sqrt{4\\frac{1}{4}(x-(-3))}+0", "displayFormula": "4\\frac{1}{4}(x-(-3))=y^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "x=-\\frac{13}{4}", "displayFormula": "x=-\\frac{13}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-3,0)", "(-\\frac{11}{4},0)" ], "pointsDecimal": [ { "fst": -3, "snd": 0 }, { "fst": -2.75, "snd": 0 } ], "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}{4}(x-(-3))=y^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -4.571428571428571, "xMax": 1.1428571428571423, "yMin": -2.8571428571428568, "yMax": 2.8571428571428568 } }, "showViewLarger": true } } }