$\\phantom{a}$
", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "This equation can be described as a
", "matrix": [0, "0", "0", 0, 0, "1"], "shuffleChoices": true, "marks": 0, "variableReplacements": [], "choices": ["$0$th degree polynomial
", "polynomial of degree $1$
", "polynomial of degree $2$
", "hyperbola
", "circle
", "polynomial of degree $\\var{n}$
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "An equation of the form $y=c_nx^n+c_{n-1}x^{n-1}+\\ldots+c_1x+c_0$ is called an $n$th degree polynomial.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "scripts": {}, "distractors": ["", "", "", "", "", ""], "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "As we move to the far left of the graph, the graph
", "matrix": "lbeh", "shuffleChoices": false, "variableReplacements": [], "choices": ["goes upwards.
", "goes downwards.
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.
\nThe leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.
\nAs we go far to the left of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "marks": 0, "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "As we move to the far right of the graph, the graph
", "matrix": "rbeh", "shuffleChoices": false, "variableReplacements": [], "choices": ["goes upwards.
", "goes downwards.
"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.
\nThe leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.
\nAs we go far to the right of the graph $x$ is negative, and so $\\simplify{{c[n]}x^{n}}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "maxMarks": 0, "marks": 0, "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "prompt": "The $y$-intercept of the graph is $y=$[[0]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{c[0]}", "minValue": "{c[0]}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "The $y$-intercept is the value of $y$ when $x=0$, that is, the value of $y$ where the graph hits the $y$-axis. To find it, substitute $x=0$ into our equation {poly}. Doing so shows that $y=\\var{c[0]}$ is the $y$-intercept.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "Given any polynomial of degree $\\var{n}$, the maximum number of $x$-intercepts in its graph is [[0]] .
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{n}", "minValue": "{n}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "The $x$-intercept is the value of $x$ when $y=0$, that is, the value of $x$ where the graph hits the $x$-axis. To find it, substitute $y=0$ into our equation:
\n{poly0}
\nThe Fundamental Theorem of Algebra says there are exactly $\\var{n}$ (complex) solutions to this equation (including multiplicity). The $x$-intercepts for our polynomial are real solutions to the above equation and therefore there are at most $\\var{n}$ $x$-intercepts.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "Given any polynomial of degree $\\var{n}$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{n-1}", "minValue": "{n-1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "A degree $n$ polynomial has at most $n-1$ bends in its graph.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "extensions": [], "statement": "You are given the equation {poly}.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"case": {"definition": "if(c[n]>0 and parity=1,1,if(c[n]>0 and parity=0,2,if(c[n]<0 and parity=1,3,4)))", "templateType": "anything", "group": "Ungrouped variables", "name": "case", "description": ""}, "parity": {"definition": "mod(n,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "parity", "description": ""}, "c": {"definition": "repeat(random(0,random(-12..12 except 0)),n)+[random(-12..12 except 0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "rbeh": {"definition": "[if(case=1 or case=2,1,0),if(case=1 or case=2,0,1)]", "templateType": "anything", "group": "Ungrouped variables", "name": "rbeh", "description": ""}, "lbeh": {"definition": "[if(case=3 or case=2,1,0),if(case=3 or case=2,0,1)]", "templateType": "anything", "group": "Ungrouped variables", "name": "lbeh", "description": ""}, "poly0": {"definition": "if(n=2,'\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2}\\$',\nif(n=3, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3}\\$',\nif(n=4, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4}\\$',\nif(n=5, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5}\\$',\nif(n=6, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6}\\$',\nif(n=7, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7}\\$',\nif(n=8, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8}\\$',\nif(n=9, '\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9}\\$',\n'\\$\\\\simplify{0={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9+{c[10]}x^{10}}\\$')))))))) ", "templateType": "anything", "group": "Ungrouped variables", "name": "poly0", "description": ""}, "poly": {"definition": "if(n=2,'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2}\\$',\nif(n=3, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3}\\$',\nif(n=4, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4}\\$',\nif(n=5, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5}\\$',\nif(n=6, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6}\\$',\nif(n=7, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7}\\$',\nif(n=8, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8}\\$',\nif(n=9, '\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9}\\$',\n'\\$\\\\simplify{y={c[0]}+{c[1]}x+{c[2]}x^2+{c[3]}x^3+{c[4]}x^4+{c[5]}x^5+{c[6]}x^6+{c[7]}x^7+{c[8]}x^8+{c[9]}x^9+{c[10]}x^{10}}\\$')))))))) ", "templateType": "anything", "group": "Ungrouped variables", "name": "poly", "description": ""}, "n": {"definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "Understanding the general facts about polynomials of degree n.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}