vertices y=2x^2+8x-34

{ "query": { "display": "vertices $$y=2x^{2}+8x-34$$", "symbolab_question": "CONIC#vertices y=2x^{2}+8x-34" }, "solution": { "level": "PERFORMED", "subject": "Geometry", "topic": "Parabola", "subTopic": "vertices", "default": "\\mathrm{Minimum}\\:(-2,-42)", "meta": { "showVerify": true } }, "methods": [ { "method": "Find vertex using polynomial form", "query": { "display": "vertex quadratic $$y=2x^{2}+8x-34$$", "symbolab_question": "vertexquadratic y=2x^{2}+8x-34" } }, { "method": "Find vertex using parabola form", "query": { "display": "vertex parabola $$y=2x^{2}+8x-34$$", "symbolab_question": "vertexparabola y=2x^{2}+8x-34" } }, { "method": "Find vertex using vertex form", "query": { "display": "vertex form $$y=2x^{2}+8x-34$$", "symbolab_question": "vertexform y=2x^{2}+8x-34" } }, { "method": "Find vertex using averaging the zeros", "query": { "display": "vertex zeros $$y=2x^{2}+8x-34$$", "symbolab_question": "vertexzeros y=2x^{2}+8x-34" } } ], "steps": { "type": "interim", "title": "Parabola vertex given $$y=2x^{2}+8x-34:{\\quad}$$Minimum $$\\left(-2,\\:-42\\right)$$", "input": "y=2x^{2}+8x-34", "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=2,\\:b=8,\\:c=-34" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{8}{2\\cdot\\:2}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=-2" }, { "type": "interim", "title": "Plug in $$x_{v}=-2\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=2\\left(-2\\right)^{2}+8\\left(-2\\right)-34", "result": "y_{v}=-42", "steps": [ { "type": "interim", "title": "Simplify $$2\\left(-2\\right)^{2}+8\\left(-2\\right)-34:{\\quad}-42$$", "input": "2\\left(-2\\right)^{2}+8\\left(-2\\right)-34", "result": "y_{v}=-42", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(-a\\right)=-a$$", "result": "=2\\left(-2\\right)^{2}-8\\cdot\\:2-34" }, { "type": "interim", "title": "$$2\\left(-2\\right)^{2}=2^{3}$$", "input": "2\\left(-2\\right)^{2}", "steps": [ { "type": "interim", "title": "$$\\left(-2\\right)^{2}=2^{2}$$", "input": "\\left(-2\\right)^{2}", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$\\left(-a\\right)^{n}=a^{n},\\:$$if $$n$$ is even", "secondary": [ "$$\\left(-2\\right)^{2}=2^{2}$$" ], "result": "=2^{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7sNuhSfBo+/I8oqMceBlhcs0ag8T1MwTer44+aCS/ZFBDeoKWfP4f0hW8hp+DjlqkWG48kfKlXwh1JXHkPaftrOeZImDuB9kLWbJJECF6RjY=" } }, { "type": "step", "result": "=2^{2}\\cdot\\:2" }, { "type": "step", "primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$", "secondary": [ "$$2\\cdot\\:2^{2}=\\:2^{1+2}$$" ], "result": "=2^{1+2}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Add the numbers: $$1+2=3$$", "result": "=2^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l59iQG3TdA68t05sDL84xt13jtrSFDx+UNsawjlOjV008YqA43ROSh5otH2dwermG+hxH/Va6lnz+40IrqhWynf3Q8V6oRlHTPRzan3GkP4=" } }, { "type": "interim", "title": "$$8\\cdot\\:2=16$$", "input": "8\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$8\\cdot\\:2=16$$", "result": "=16" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7y2h4Tex+34gTCvdQ87mLo913jtrSFDx+UNsawjlOjV1pDjGlV0QXxhP2pBsH2aZoKMGNP280hCjL6civ0YbL5CaJYndagF0H9k5ZrjA1ruE=" } }, { "type": "step", "result": "=2^{3}-16-34" }, { "type": "step", "primary": "Subtract the numbers: $$-16-34=-50$$", "result": "=2^{3}-50" }, { "type": "step", "primary": "$$2^{3}=8$$", "result": "=8-50" }, { "type": "step", "primary": "Subtract the numbers: $$8-50=-42$$", "result": "=-42" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ldZ081ez390fp1Qm4Bb9d3kzpThuKjSq5L9U5lDMpETMwViaLUXkeD+JukROhWdj5oy1I6/7FjgnyaRnyblJtJjcBIL5pmo83UMFZRSzJsW6jjdSfGZUQ7LD4AUzZai+d4Qsg9+S1e67pedWvtepHw==" } } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(-2,\\:-42\\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/>$$a=2$$", "result": "\\mathrm{Minimum}\\:\\left(-2,\\:-42\\right)" } ], "meta": { "solvingClass": "Parabola" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "funcsToDraw": { "funcs": [ { "evalFormula": "y=\\frac{(x-(-2))^{2}}{4\\frac{1}{8}}-42", "displayFormula": "4\\frac{1}{8}(y-(-42))=(x-(-2))^{2}", "attributes": { "color": "PURPLE", "lineType": "NORMAL", "isAsymptote": false } }, { "evalFormula": "y=-\\frac{337}{8}", "displayFormula": "y=-\\frac{337}{8}", "attributes": { "color": "GRAY", "lineType": "NORMAL", "labels": [ "\\mathrm{directrix}" ], "isAsymptote": false } } ] }, "pointsToDraw": { "pointsLatex": [ "(-2,-42)", "(-2,-\\frac{335}{8})" ], "pointsDecimal": [ { "fst": -2, "snd": -42 }, { "fst": -2, "snd": -41.875 } ], "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}{8}(y-(-42))=(x-(-2))^{2}", "paramsLatex": [], "paramsReplacementsLatex": [] } ], "localBoundingBox": { "xMin": -31.428571428571427, "xMax": 29.28571428571428, "yMin": -48.57142857142857, "yMax": 12.142857142857139 } }, "showViewLarger": true } }, "meta": { "showVerify": true } }