range of x^2-2x-18

{ "query": { "display": "range $$x^{2}-2x-18$$", "symbolab_question": "FUNCTION#range x^{2}-2x-18" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "range", "default": "f(x)\\ge -19", "interval": "[-19,\\infty )", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Range of $$x^{2}-2x-18:{\\quad}f\\left(x\\right)\\ge\\:-19$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "interim", "title": "Vertex of $$x^{2}-2x-18:{\\quad}$$Minimum $$\\left(1,\\:-19\\right)$$", "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=-2,\\:c=-18" }, { "type": "step", "primary": "$$x_{v}=-\\frac{b}{2a}$$", "result": "x_{v}=-\\frac{\\left(-2\\right)}{2\\cdot\\:1}" }, { "type": "step", "primary": "Simplify", "result": "x_{v}=1" }, { "type": "interim", "title": "Plug in $$x_{v}=1\\:$$to find the $$y_{v}\\:$$value", "input": "y_{v}=1^{2}-2\\cdot\\:1-18", "result": "y_{v}=-19", "steps": [ { "type": "step", "primary": "Simplify", "result": "y_{v}=-19" } ], "meta": { "interimType": "Plug In Value 2Eq" } }, { "type": "step", "primary": "Therefore the parabola vertex is", "result": "\\left(1,\\:-19\\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(1,\\:-19\\right)" } ], "meta": { "solvingClass": "Function Vertex", "interimType": "Range Parabola Find Vertex 1Eq" } }, { "type": "step", "primary": "For a parabola $$ax^2+bx+c\\:$$with Vertex $$\\left(x_v,\\:y_v\\right)$$<br/>$$\\quad$$If $$a<0\\:$$the range is $$f\\left(x\\right)\\le\\:y_v$$<br/>$$\\quad$$If $$a>0\\:$$the range is $$f\\left(x\\right)\\ge\\:y_v$$<br/>$$a=1,\\:$$Vertex $$\\left(x_v,\\:y_v\\right)=\\left(1,\\:-19\\right)$$", "result": "f\\left(x\\right)\\ge\\:-19" } ], "meta": { "solvingClass": "Function Range" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}-2x-18" }, "showViewLarger": true } }, "meta": { "showVerify": true } }