intercepts of f(x)=(x-4)^2

{ "query": { "display": "intercepts $$f\\left(x\\right)=\\left(x-4\\right)^{2}$$", "symbolab_question": "CONIC#intercepts f(x)=(x-4)^{2}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{X\\:Intercepts}: (4,0),\\mathrm{Y\\:Intercepts}: (0,16)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$\\left(x-4\\right)^{2}:\\quad\\:$$X Intercepts$$:\\:\\left(4,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:16\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$\\left(x-4\\right)^{2}:{\\quad}\\left(4,\\:0\\right)$$", "input": "\\left(x-4\\right)^{2}", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$\\left(x-4\\right)^{2}=0:{\\quad}x=4$$", "input": "\\left(x-4\\right)^{2}=0", "steps": [ { "type": "step", "primary": "Using the Zero Factor Principle:$$\\quad$$ If $$ab=0\\:$$then $$a=0\\:$$or $$b=0$$" }, { "type": "interim", "title": "Solve $$x-4=0:{\\quad}x=4$$", "input": "x-4=0", "steps": [ { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "x-4=0", "result": "x=4", "steps": [ { "type": "step", "primary": "Add $$4$$ to both sides", "result": "x-4+4=0+4" }, { "type": "step", "primary": "Simplify", "result": "x=4" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "primary": "The solution to the quadratic equation is:", "result": "x=4" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(4,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7Pse0nBGqHVGUbc/7OZIemr3g2F2yQn3vCs6qoxcWnKGqGgBg7+Qqjt3KiwKdNp2BLWFmDzgQyN3TmtLk4lh8AZnnDDyRtsdwXnB+1ZL6pkNA==" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$\\left(x-4\\right)^{2}:{\\quad}\\left(0,\\:16\\right)$$", "input": "\\left(x-4\\right)^{2}", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=\\left(0-4\\right)^{2}:{\\quad}y=16$$", "input": "y=\\left(0-4\\right)^{2}", "steps": [ { "type": "interim", "title": "Simplify $$\\left(0-4\\right)^{2}:{\\quad}16$$", "input": "\\left(0-4\\right)^{2}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$0-4=-4$$", "result": "=\\left(-4\\right)^{2}" }, { "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}" }, { "type": "step", "primary": "$$4^{2}=16$$", "result": "=16" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rsLKCW/yKKFM+tShRnQiVN13jtrSFDx+UNsawjlOjV1pDjGlV0QXxhP2pBsH2aZo8LfSxJ+0AgVLpCSnLX0iSqZGbEKkvNH3vNWpvWJeVIx5Mk0scJXhFSCpBf+bKNBJ" } }, { "type": "step", "result": "y=16" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:16\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcflz/9QAa8cUih2jlLdjjE5QfXtjcKfn9o/LtdD1OV+zPmDlwPZY/s+3QwhHk6uKuROtFqYeaNaWjmLLIEQ4Mm7uf7D6sQaozkrtEMXNocgjNw==" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(4,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:16\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "(x-4)^{2}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }