// Numbas version: exam_results_page_options {"name": "Graphing: polynomials", "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "shuffleQuestions": true, "percentPass": 0, "duration": 0, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-shuffled", "pickQuestions": 0, "questions": [{"name": "Graphing: linear", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["a", "b", "lleading", "rleading", "xints"], "tags": ["graphing", "linear", "polynomials", "sketching", "straight lines"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "displayColumns": 0, "prompt": "

This equation, or its graph, can be described as a

", "matrix": ["1", "0", "0", 0, 0, 0], "shuffleChoices": true, "maxMarks": 0, "variableReplacements": [], "choices": ["

straight line

", "

parabola/quadratic

", "

cubic

", "

hyperbola

", "

circle

", "

quartic

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An equation of the form $y=ax+b$ is known as a linear equation, and its graph is a straight line.

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As we move to the far left of the graph, the graph

", "matrix": "lleading", "shuffleChoices": false, "maxMarks": 0, "variableReplacements": [], "choices": ["

goes upwards.

", "

goes downwards.

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What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. In our polynomial the leading term is $\\simplify[all,fractionNumbers]{{a}x}$.

As we go far to the left of the graph $x$ is negative, and so $\\simplify[all,fractionNumbers]{{a}x}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"stepsPenalty": "1", "displayColumns": 0, "prompt": "

As we move to the far right of the graph, the graph

", "matrix": "rleading", "shuffleChoices": false, "maxMarks": 0, "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. In our polynomial the leading term is $\\simplify[all,fractionNumbers]{{a}x}$.

As we go far to the right of the graph $x$ is positive, and so $\\simplify[all,fractionNumbers]{{a}x}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

The $y$-intercept of the graph is $y=$[[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[y=\\simplify[unitFactor,basic,fractionNumbers]{{a}0+{b}}=\\var{b}.\\]

The set of $x$-intercepts of the graph would be [[0]].

Note: If there are no intercepts, enter set()

If there is only one intercept, say $x=5$, enter set(5)

If there are two intercepts, say $x=-2$ and $x=1.5$, enter set(-2,1.5)

If there are three intercepts, say $x=-2$, $x=1.5$ and $x=5$, enter set(-2,1.5,5)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[0=\\simplify[all,fractionNumbers]{{a}x+{b}} \\]

Solving this equation tells us that the $x$-intercept is $x=\\simplify[all, fractionNumbers]{{-b}/{a}}$.

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Given the degree of a polynomial is $1$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

A degree $n$ polynomial has at most $n-1$ bends in its graph.

You are given the equation $y=\\simplify[all,fractionNumbers]{{a}x+{b}}$.

This equation, or its graph, can be described as a

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straight line

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parabola/quadratic

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cubic

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hyperbola

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circle

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quartic

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An equation of the form $y=ax^2+bx+c$ is a quadratic and the graph of it is called a parabola.

The graph of this equation

", "matrix": "concave", "shuffleChoices": false, "maxMarks": 0, "variableReplacements": [], "choices": ["

opens upwards (is concave up)

", "

opens downwards (is concave down)

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The number in front of $x^2$ is called the coefficient of $x^2$ or the leading coefficient because it belongs to the term with the highest power.

If this coefficient is postive then the parabola will be concave up, or 'open upwards', or in other words, look like a smiley face!

If this coefficient is negative then the parabola will be concave down, or 'open downwards', or in other words, look like a sad face!

The $y$-intercept of the graph is $y=$[[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[y=\\simplify[unitFactor,basic,fractionNumbers]{{a}0^2+{b}0+{c}}=\\var{c}.\\]

The axis of symmetry of the graph of this function is $x=$[[0]].

The vertex of the graph of this function is at $\\large($ [[0]], [[1]] $\\large)$.

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

The axis of symmetry of a parabola is the line along which you could reflect the parabola and it would look exactly the same. For the quadratic $y=ax^2+bx+c$, the equation of the axis of symmetry is \\[x=-\\dfrac{b}{2a}.\\] Using our given equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we get $x=\\var{axis_x}$.

The vertex, or the turning point, or the stationary point of the parabola is on this line, so it's $x$ coordinate must be $\\var{axis_x}$ and its $y$ coordinate must be the $y$ value that corresponds to this $x$ value after susbstituting it into the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$. That is \\[y=\\simplify[unitFactor,basic,fractionNumbers]{{a}({axis_x})^2+{b}({axis_x})+{c}}=\\var{axis_y}.\\]

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The set of $x$-intercepts of the graph would be [[0]].

Note: If there are no intercepts, enter set()

If there is only one intercept, say $x=5$, enter set(5)

If there are two intercepts, say $x=-2$ and $x=1.5$, enter set(-2,1.5)

If there are three intercepts, say $x=-2$, $x=1.5$ and $x=5$, enter set(-2,1.5,5)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

For the quadratic $y=ax^2+bx+c$, the $x$-intercepts (if they exist) are given by \\[x=\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}.\\]

Notice how the first term gives the axis of symmetry and the second term says how far away from the axis of symmetry to go in both directions!

If $b^2-4ac>0$ then there will be two $x$-intercepts.
If $b^2-4ac=0$ then there will be one $x$-intercept and it will coincide with the axis of symmetry.
If $b^2-4ac<0$ then there will be no $x$-intercepts since the square root of a negative number is not a real number.

For the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we have $b^2-4ac=\\simplify[all,fractionNumbers]{{b}^2-4{a}{c}}=\\var{disc}$ and so there are no $x$-intercepts. is one $x$-intercept: are two $x$-intercepts:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$x$	$=$	$\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}$

	$=$	$\\simplify[basic,unitFactor,fractionNumbers]{{axis_x}}\\pm\\simplify[basic,unitFactor,fractionNumbers]{sqrt{{disc}}/({2*a})}$

	$=$	$\\var{axis_x}$ $\\var{xint0}, \\, \\var{xint1}$

Given the degree of a polynomial is $2$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

A degree $n$ polynomial has at most $n-1$ bends in its graph.

You are given the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "if(switch=0, thres,if(switch<0,(b^2+s^2)/(4a),(b^2-s^2)/(4a)))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "-2*a*axis_x", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "xint0": {"definition": "axis_x-s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint0", "description": ""}, "concave": {"definition": "[if(a>0,1,0),if(a<0,1,0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "concave", "description": ""}, "switch": {"definition": "random(-1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "switch", "description": ""}, "xints": {"definition": "if(switch=-1,set() ,if(switch=0,set(axis_x),set(axis_x-s/(2*a),axis_x+s/(2*a))))", "templateType": "anything", "group": "Ungrouped variables", "name": "xints", "description": ""}, "s": {"definition": "random(map(n^2,n,1..12))*2*a", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "disc": {"definition": "b^2-4*a*c", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "axis_x": {"definition": "random(-4..4#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_x", "description": ""}, "axis_y": {"definition": "a*axis_x^2+b*axis_x+c", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_y", "description": ""}, "xint1": {"definition": "axis_x+s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint1", "description": ""}, "thres": {"definition": "b^2/(4a)", "templateType": "anything", "group": "Ungrouped variables", "name": "thres", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Graphing: cubic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["switch", "axis_x", "a", "b", "thres", "s", "c", "lleading", "rleading", "axis_y", "xints", "disc", "xint0", "xint1", "d", "yint"], "tags": ["cubic", "graphing", "polynomial", "Polynomial", "polynomials", "sketching"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "displayColumns": 0, "prompt": "

This equation, or its graph, can be described as a

", "matrix": [0, "0", "1", 0, 0, 0], "shuffleChoices": true, "maxMarks": 0, "variableReplacements": [], "choices": ["

straight line

", "

parabola/quadratic

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cubic

", "

hyperbola

", "

circle

", "

quartic

"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

An equation of the form $y=ax^3+bx^2+cx+d$ is known as a cubic, or a cubic polynomial. If we expand $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we will see it is a cubic.

As we move to the far left of the graph, the graph

", "matrix": "lleading", "shuffleChoices": false, "maxMarks": 0, "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial.

By expanding $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^3}$.

As we go far to the left of the graph $x$ is negative, and so $\\simplify[all,fractionNumbers]{{a}x^3}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

As we move to the far right of the graph, the graph

", "matrix": "rleading", "shuffleChoices": false, "maxMarks": 0, "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial.

By expanding $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^3}$.

As we go far to the right of the graph $x$ is positive, and so $\\simplify[all,fractionNumbers]{{a}x^3}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

The $y$-intercept of the graph is $y=$[[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[y=\\simplify[unitFactor,basic,fractionNumbers]{(0-{d})({a}0^2+{b}0+{c})}=\\var{yint}.\\]

The set of $x$-intercepts of the graph would be [[0]].

Note: If there are no intercepts, enter set()

If there is only one intercept, say $x=5$, enter set(5)

If there are two intercepts, say $x=-2$ and $x=1.5$, enter set(-2,1.5)

If there are three intercepts, say $x=-2$, $x=1.5$ and $x=5$, enter set(-2,1.5,5)

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[0=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})} \\]

Recall, if a product is zero then one of the factors must be zero, therefore

\\[\\simplify[all,fractionNumbers]{x-{d}=0}\\quad \\text{or}\\quad\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}=0}.\\]

Solving the first equation says that one of the $x$-intercepts is $x=\\var{d}$.

For the second equation we will use the quadratic formula. Recall for $ax^2+bx+c=0$, the solutions (if they exist) are given by \\[x=\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}.\\]

If $b^2-4ac>0$ then there will be two more $x$-intercepts.
If $b^2-4ac=0$ then there will be one more $x$-intercept.
If $b^2-4ac<0$ then there will be no more $x$-intercepts since the square root of a negative number is not a real number.

For the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we have $b^2-4ac=\\simplify[basic,unitFactor,fractionNumbers]{{b}^2-4{a}{c}}=\\var{disc}$ and so there are no more $x$-intercepts. is one more $x$-intercept: are two more $x$-intercepts:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$x$	$=$	$\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}$

	$=$	$\\simplify[basic,unitFactor,fractionNumbers]{{axis_x}}\\pm\\simplify[basic,unitFactor,fractionNumbers]{sqrt{{disc}}/({2*a})}$

	$=$	$\\var{axis_x}$ $\\var{xint0}, \\, \\var{xint1}$

Given the degree of a polynomial is $3$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

A degree $n$ polynomial has at most $n-1$ bends in its graph.

You are given the equation $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "if(switch=0, thres,if(switch<0,(b^2+s^2)/(4a),(b^2-s^2)/(4a)))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "-2*a*axis_x", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-10..10 except [xint0, xint0])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "yint": {"definition": "-d*c", "templateType": "anything", "group": "Ungrouped variables", "name": "yint", "description": ""}, "rleading": {"definition": "[if(a>0,1,0),if(a<0,1,0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "rleading", "description": ""}, "xint0": {"definition": "axis_x-s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint0", "description": ""}, "switch": {"definition": "random(-1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "switch", "description": ""}, "xints": {"definition": "if(switch=-1,set(d) ,if(switch=0,set(d,axis_x),set(d,axis_x-s/(2*a),axis_x+s/(2*a))))", "templateType": "anything", "group": "Ungrouped variables", "name": "xints", "description": ""}, "s": {"definition": "random(map(n^2,n,1..12))*2*a", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "disc": {"definition": "b^2-4*a*c", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "axis_x": {"definition": "random(-4..4#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_x", "description": ""}, "axis_y": {"definition": "a*axis_x^2+b*axis_x+c", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_y", "description": ""}, "xint1": {"definition": "axis_x+s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint1", "description": ""}, "thres": {"definition": "b^2/(4a)", "templateType": "anything", "group": "Ungrouped variables", "name": "thres", "description": ""}, "lleading": {"definition": "[if(a<0,1,0),if(a>0,1,0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "lleading", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Graphing: quartic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["switch", "axis_x", "a", "b", "thres", "s", "c", "lleading", "rleading", "axis_y", "xints", "disc", "xint0", "xint1", "d", "yint", "ee"], "tags": ["graphing", "polynomial", "Polynomial", "polynomials", "quartic", "sketching"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "displayColumns": 0, "prompt": "

This equation, or its graph, can be described as a

", "matrix": [0, "0", "0", 0, 0, "1"], "shuffleChoices": true, "maxMarks": 0, "variableReplacements": [], "choices": ["

straight line

", "

parabola/quadratic

", "

cubic

", "

hyperbola

", "

circle

", "

quartic

"], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

An equation of the form $y=ax^4+bx^3+cx^2+dx+e$ is known as a quartic, or a quartic polynomial. If we expand $y=\\simplify[all,fractionNumbers]{(x-{ee})(x-{d})({a}x^2+{b}x+{c})}$ we will see it is a quartic.

As we move to the far left of the graph, the graph

", "matrix": "lleading", "shuffleChoices": false, "maxMarks": 0, "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial.

By expanding $y=\\simplify[all,fractionNumbers]{(x-{ee})(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^4}$.

As we go far to the left of the graph $x$ is negative, and so $\\simplify[all,fractionNumbers]{{a}x^4}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

As we move to the far right of the graph, the graph

", "matrix": "rleading", "shuffleChoices": false, "maxMarks": 0, "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial.

By expanding $y=\\simplify[all,fractionNumbers]{(x-{ee})(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^4}$.

As we go far to the right of the graph $x$ is positive, and so $\\simplify[all,fractionNumbers]{{a}x^4}$ is negative. That is, the graph goes downwards. is positive. That is, the graph goes upwards.

The $y$-intercept of the graph is $y=$[[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[y=\\simplify[unitFactor,basic,fractionNumbers]{(0-{ee})(0-{d})({a}0^2+{b}0+{c})}=\\var{yint}.\\]

The set of $x$-intercepts of the graph would be [[0]].

Note: If there are no intercepts, enter set()

If there is only one intercept, say $x=5$, enter set(5)

If there are two intercepts, say $x=-2$ and $x=1.5$, enter set(-2,1.5)

If there are three intercepts, say $x=-2$, $x=1.5$ and $x=5$, enter set(-2,1.5,5)

etc

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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:

\\[0=\\simplify[all,fractionNumbers]{(x-{ee})(x-{d})({a}x^2+{b}x+{c})} \\]

Recall, if a product is zero then one of the factors must be zero, therefore

\\[\\simplify[all,fractionNumbers]{x-{ee}=0}\\quad \\textrm{or}\\quad\\simplify[all,fractionNumbers]{x-{d}=0}\\quad \\text{or}\\quad\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}=0}.\\]

Solving the first equation says that one of the $x$-intercepts is $x=\\var{ee}$.

Solving the second equation says that one of the $x$-intercepts is $x=\\var{d}$.

For the third equation we will use the quadratic formula. Recall for $ax^2+bx+c=0$, the solutions (if they exist) are given by \\[x=\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}.\\]

If $b^2-4ac>0$ then there will be two more $x$-intercepts.
If $b^2-4ac=0$ then there will be one more $x$-intercept.
If $b^2-4ac<0$ then there will be no more $x$-intercepts since the square root of a negative number is not a real number.

For the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we have $b^2-4ac=\\simplify[all,fractionNumbers]{{b}^2-4{a}{c}}=\\var{disc}$ and so there are no more $x$-intercepts. is one more $x$-intercept: are two more $x$-intercepts:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$x$	$=$	$\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}$

	$=$	$\\simplify[basic,unitFactor,fractionNumbers]{{axis_x}}\\pm\\simplify[basic,unitFactor,fractionNumbers]{sqrt{{disc}}/({2*a})}$

	$=$	$\\var{axis_x}$ $\\var{xint0}, \\, \\var{xint1}$

Given the degree of a polynomial is $4$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "

A degree $n$ polynomial has at most $n-1$ bends in its graph.

You are given the equation $y=\\simplify[all,fractionNumbers]{(x-{ee})(x-{d})({a}x^2+{b}x+{c})}$.

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "if(switch=0, thres,if(switch<0,(b^2+s^2)/(4a),(b^2-s^2)/(4a)))", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "-2*a*axis_x", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-6..6 except [xint0, xint0])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "yint": {"definition": "ee*d*c", "templateType": "anything", "group": "Ungrouped variables", "name": "yint", "description": ""}, "rleading": {"definition": "[if(a>0,1,0),if(a<0,1,0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "rleading", "description": ""}, "xint0": {"definition": "axis_x-s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint0", "description": ""}, "lleading": {"definition": "[if(a>0,1,0),if(a<0,1,0)]", "templateType": "anything", "group": "Ungrouped variables", "name": "lleading", "description": ""}, "xints": {"definition": "if(switch=-1,set(d,ee) ,if(switch=0,set(d,ee,axis_x),set(d,ee,axis_x-s/(2*a),axis_x+s/(2*a))))", "templateType": "anything", "group": "Ungrouped variables", "name": "xints", "description": ""}, "ee": {"definition": "random(-5..5 except [xint0, xint0])", "templateType": "anything", "group": "Ungrouped variables", "name": "ee", "description": ""}, "s": {"definition": "random(map(n^2,n,1..12))*2*a", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "disc": {"definition": "b^2-4*a*c", "templateType": "anything", "group": "Ungrouped variables", "name": "disc", "description": ""}, "axis_x": {"definition": "random(-4..4#0.5)", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_x", "description": ""}, "axis_y": {"definition": "a*axis_x^2+b*axis_x+c", "templateType": "anything", "group": "Ungrouped variables", "name": "axis_y", "description": ""}, "xint1": {"definition": "axis_x+s/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "xint1", "description": ""}, "thres": {"definition": "b^2/(4a)", "templateType": "anything", "group": "Ungrouped variables", "name": "thres", "description": ""}, "switch": {"definition": "random(-1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "switch", "description": ""}}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Graphing: nth degree polynomial", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "functions": {}, "ungrouped_variables": ["n", "c", "parity", "poly", "case", "lbeh", "rbeh", "poly0"], "tags": ["graphing", "polynomial", "polynomials", "sketching"], "advice": "

$\\phantom{a}$

", "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.

As 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.

", "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.

The leading term (the term that includes the highest power) determines the long term behaviour of a polynomial. For our polynomial this is $\\simplify{{c[n]}x^{n}}$.

As 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.

", "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.

", "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]] .

", "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:

{poly0}

The 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.

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.

You are given the equation {poly}.

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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/"}], "extensions": [], "custom_part_types": [], "resources": []}