shift 4sin(3θ-1/3 pi)+1

{ "query": { "display": "shift $$4\\sin\\left(3θ-\\frac{1}{3}π\\right)+1$$", "symbolab_question": "FUNCTION#shift 4\\sin(3θ-\\frac{1}{3}π)+1" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "shift", "default": "\\mathrm{Phase}:\\frac{π}{9},\\:\\mathrm{Vertical}:1", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Shift of $$4\\sin\\left(3θ-\\frac{1}{3}π\\right)+1:{\\quad}$$Phase$$:\\frac{π}{9},\\:\\:$$Vertical$$:1$$", "steps": [ { "type": "definition", "title": "Shift definition", "text": "For $$f\\left(x\\right)=A{\\cdot}g\\left(Bx-C\\right)+D$$, where $$g\\left(x\\right)$$ is one of the basic trig functions, <br/>$$\\frac{C}{B}$$ is phase shift<br/>$$D$$ is vertical shift" }, { "type": "interim", "title": "Simplify $$4\\sin\\left(3θ-\\frac{1}{3}π\\right)+1:{\\quad}4\\sin\\left(3θ-\\frac{π}{3}\\right)+1$$", "input": "4\\sin\\left(3θ-\\frac{1}{3}π\\right)+1", "steps": [ { "type": "interim", "title": "$$\\frac{1}{3}π=\\frac{π}{3}$$", "input": "\\frac{1}{3}π", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1π}{3}" }, { "type": "step", "primary": "Multiply: $$1π=π$$", "result": "=\\frac{π}{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s772I7EUUf95BSZLtUnBRqOACWKUbvV6WK3fDUgFtg3Q/ysS6ztOAKKFAEXmiY4WQmAonq/n3jQ0ddQ1SaaPv11c1bIZxfodm3UsZcfZAZr4vkhcFupvBZeQVoyGlC+ZKTuNtnTCAhUVHtS+6YazBAQQ==" } }, { "type": "step", "result": "=4\\sin\\left(3θ-\\frac{π}{3}\\right)+1" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s76cV9vr+rRKMYwsvGWq+AKkWjrDkBItR8MRDcZM/LONVV00rpv8+ZC6TM10tVCSHslFSVkxrxkAklcCEjrnolWf3r1JJoufY0TjRXsw2kvZMj2HLr5XkgDz1O/djA+zZzHjb2+5NLFZrsH9fcPWg/TQKMiz5PaV4XkV7s9WTwtTyhhLKIyifLo0PBaGhQutFIRUAEcmWNuAmUYsCX+4H8TA==" } }, { "type": "step", "primary": "$$f\\left(x\\right)=4\\sin\\left(3θ-\\frac{π}{3}\\right)+1$$<br/>$$g\\left(Bx-C\\right)=\\sin\\left(3θ-\\frac{π}{3}\\right),\\:B=3,\\:C=\\frac{π}{3},\\:D=1$$<br/>Therefore, the phase shift $$\\frac{C}{B}$$ is $$\\frac{\\frac{π}{3}}{3}$$", "result": "\\frac{π}{9}" }, { "type": "step", "primary": "The vertical shift $$D$$ is $$1$$", "result": "1" } ], "meta": { "solvingClass": "Function Shift" } }, "plot_output": { "meta": { "plotInfo": { "variable": "θ", "plotRequest": "4\\sin(3θ-\\frac{π}{3})+1" }, "showViewLarger": true } }, "meta": { "showVerify": true } }