vertices f(x)=x^2+16x+63

{ "query": { "display": "vertices $$f\\left(x\\right)=x^{2}+16x+63$$", "symbolab_question": "CONIC#vertices f(x)=x^{2}+16x+63" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-8,-1)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=x^{2}+16x+63$$", "symbolab_question": "vertexquadratic y=x^{2}+16x+63" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=x^{2}+16x+63$$", "symbolab_question": "vertexparabola y=x^{2}+16x+63" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=x^{2}+16x+63$$", "symbolab_question": "vertexform y=x^{2}+16x+63" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=x^{2}+16x+63$$", "symbolab_question": "vertexzeros y=x^{2}+16x+63" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=x^{2}+16x+63:{\\quad}$$Minimum $$\\left(-8,\\:-1\\right)$$", "input": "y=x^{2}+16x+63", "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=16,\\:c=63" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{16}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-8" }, { "type": "interim", "title": "Plug in $$x_{v}=-8\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=\\left(-8\\right)^{2}+16\\left(-8\\right)+63", "result": "y_{v}=-1", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-8\\right)^{2}+16\\left(-8\\right)+63:{\\quad}-1$$", "input": "\\left(-8\\right)^{2}+16\\left(-8\\right)+63", "result": "y_{v}=-1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\left(-8\\right)^{2}-16\\cdot\\:8+63" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-8\\right)^{2}=8^{2}$$" ], "result": "=8^{2}-16\\cdot\\:8+63" }, { "type": "step", "primary": "Multiply the numbers: $$16\\cdot\\:8=128$$", "result": "=8^{2}-128+63" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-128+63=-65$$", "result": "=8^{2}-65" }, { "type": "step", "primary": "$$8^{2}=64$$", "result": "=64-65" }, { "type": "step", "primary": "Subtract the numbers: $$64-65=-1$$", "result": "=-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pZTAwIVAcfXEoSIq6Ye7E243Hsnxtr/cAxSgks+jFq7MwViaLUXkeD+JukROhWdjDZ9XI/x3sTbJ8I2f4Mzi5e5AIz++qluupTlLFEcE9J1hx6n3zRk7fdn+bvmbJ0P3f959s9+ZlS07KBbNsV9AWA==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-8,\\:-1\\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(-8,\\:-1\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-8))^{2}}{4\\frac{1}{4}}-1", "displayFormula": "4\\frac{1}{4}(y-(-1))=(x-(-8))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{5}{4}", "displayFormula": "y=-\\frac{5}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-8,-1)", "(-8,-\\frac{3}{4})" ], "pointsDecimal": [ { "fst": -8, "snd": -1 }, { "fst": -8, "snd": -0.75 } ], "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}(y-(-1))=(x-(-8))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -10.285714285714285, "xMax": 2.571428571428572, "yMin": -7.271428571428571, "yMax": 5.585714285714286 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }