vertices y=+x^2+8x+9

{ "query": { "display": "vertices $$y=+x^{2}+8x+9$$", "symbolab_question": "CONIC#vertices y=+x^{2}+8x+9" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-4,-7)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=+x^{2}+8x+9$$", "symbolab_question": "vertexquadratic y=+x^{2}+8x+9" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=+x^{2}+8x+9$$", "symbolab_question": "vertexparabola y=+x^{2}+8x+9" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=+x^{2}+8x+9$$", "symbolab_question": "vertexform y=+x^{2}+8x+9" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=+x^{2}+8x+9$$", "symbolab_question": "vertexzeros y=+x^{2}+8x+9" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=+x^{2}+8x+9:{\\quad}$$Minimum $$\\left(-4,\\:-7\\right)$$", "input": "y=x^{2}+8x+9", "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=8,\\:c=9" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{8}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-4" }, { "type": "interim", "title": "Plug in $$x_{v}=-4\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=\\left(-4\\right)^{2}+8\\left(-4\\right)+9", "result": "y_{v}=-7", "steps": [ { "type": "interim", "title": "Simplify $$\\left(-4\\right)^{2}+8\\left(-4\\right)+9:{\\quad}-7$$", "input": "\\left(-4\\right)^{2}+8\\left(-4\\right)+9", "result": "y_{v}=-7", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=\\left(-4\\right)^{2}-8\\cdot\\:4+9" }, { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-4\\right)^{2}=4^{2}$$" ], "result": "=4^{2}-8\\cdot\\:4+9" }, { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:4=32$$", "result": "=4^{2}-32+9" }, { "type": "step", "primary": "Add/Subtract the numbers: $$-32+9=-23$$", "result": "=4^{2}-23" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16-23" }, { "type": "step", "primary": "Subtract the numbers: $$16-23=-7$$", "result": "=-7" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hxGxpuiRjPRXhPKIGx+w6Lx49UHm/jSOfNcUJbiTFL95tMpJTBBccUWkSyvMe1SpDIw7UgN7P60YbOhZnn/Zs3KF3u2OIb4bFA3EO8aRlSVJEc8cpYS3hillUmBU+3GJfgwK8cFA4k97q3o95lS0fw==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-4,\\:-7\\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(-4,\\:-7\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-4))^{2}}{4\\frac{1}{4}}-7", "displayFormula": "4\\frac{1}{4}(y-(-7))=(x-(-4))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{29}{4}", "displayFormula": "y=-\\frac{29}{4}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-4,-7)", "(-4,-\\frac{27}{4})" ], "pointsDecimal": [ { "fst": -4, "snd": -7 }, { "fst": -4, "snd": -6.75 } ], "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-(-7))=(x-(-4))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -7.857142857142856, "xMax": 3.571428571428571, "yMin": -9.142857142857142, "yMax": 2.2857142857142847 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }