// Numbas version: exam_results_page_options {"name": "nonlinear relationships", "metadata": {"description": "

This is a set of questions for students to practice identifying parabolas, hyperbolas and exponentials.

There are also a few questions asking students to draw graphs, and to evaluate the curves at specific points.

10 questions are selected from a larger pool.

In the first question students are asked to identify the type of a graph.

In the second question students are asked to identify the type of an equation.

Then next 6 questions are basic questions about evaluating points on a curve or matching curves and equations.

The last 2 questions are applications - e.g. compound interest, displayed as an equation, a table or a graph.

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Students are shown an exponential curve and asked to identify it as either linear, parabolic, hyperbolic or exponential.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

{geogebra_applet('https://www.geogebra.org/m/a2bvpj8m',defs)}

", "advice": "

This curve is an exponential curve with an asymptote at $y = \\var{c}$.

This is a

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Students are shown a hyperbola and asked to identify it as such.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

{geogebra_applet('https://www.geogebra.org/m/fbzq6qkz',defs)}

", "advice": "

The graph is a hyperbola.

This is a

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["parabola", "hyperbola", "exponential", "none of these"], "matrix": ["0", "1", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "parabolic curve", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Can you identify a parabola? Students are shown a parabola and asked to identify it as such.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

{geogebra_applet('https://www.geogebra.org/m/wu4fe7gg',defs)}

", "advice": "

This is a quadratic equation, i.e. a parabola.

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This is a

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["parabola", "hyperbola", "exponential", "none of these"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "random-subset", "pickQuestions": 1, "questionNames": ["", "", "", ""], "questions": [{"name": "parabola equation", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given the equation of a parabola and asked to identify its graph. They are given the parabolic curve, an exponential, a hyperbola and a line to choose from.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of the following graphs represents $y = \\var{pchoicedispa} \\text{x}^2 \\var{choosesignb} \\var{choosesign}$?

", "advice": "

The equation given is for a parabola (quadratic).

", "rulesets": {}, "variables": {"pa": {"name": "pa", "group": "Ungrouped variables", "definition": "pachoices[pchoosea]", "description": "", "templateType": "anything"}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "random(-3,3)", "description": "", "templateType": "anything"}, "pdefs": {"name": "pdefs", "group": "Ungrouped variables", "definition": "[\n ['a',pa],['b',pb],['c',pc]\n ]", "description": "", "templateType": "anything"}, "ea": {"name": "ea", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything"}, "eb": {"name": "eb", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything"}, "ec": {"name": "ec", "group": "Ungrouped variables", "definition": "random(-2..3)", "description": "", "templateType": "anything"}, "ek": {"name": "ek", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "edefs": {"name": "edefs", "group": "Ungrouped variables", "definition": "[\n ['a',ea],['b',eb],['c',ec],['k',ek]\n ]", "description": "", "templateType": "anything"}, "ha": {"name": "ha", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3,4,5)", "description": "", "templateType": "anything"}, "hc": {"name": "hc", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "hdefs": {"name": "hdefs", "group": "Ungrouped variables", "definition": "[\n ['a',ha],['c',hc]\n ]", "description": "", "templateType": "anything"}, "ldefs": {"name": "ldefs", "group": "Ungrouped variables", "definition": "[\n ['a',pa],['b',pb]\n ]", "description": "", "templateType": "anything"}, "sign": {"name": "sign", "group": "Ungrouped variables", "definition": "['{pc}','','+ {pc}']", "description": "", "templateType": "anything"}, "choosesign": {"name": "choosesign", "group": "Ungrouped variables", "definition": "sign[sgn(pc)+1]", "description": "", "templateType": "anything"}, "signb": {"name": "signb", "group": "Ungrouped variables", "definition": "['-3x','-2x','-1x','','+ x','+ 2x','+ 3x']", "description": "", "templateType": "anything"}, "choosesignb": {"name": "choosesignb", "group": "Ungrouped variables", "definition": "signb[pb+3]", "description": "", "templateType": "anything"}, "pchoosea": {"name": "pchoosea", "group": "Ungrouped variables", "definition": "random(0..5)", "description": "", "templateType": "anything"}, "pachoices": {"name": "pachoices", "group": "Ungrouped variables", "definition": "[-3,-2,-1,1,2,3]", "description": "", "templateType": "anything"}, "hchoosea": {"name": "hchoosea", "group": "Ungrouped variables", "definition": "random(0..7)", "description": "", "templateType": "anything"}, "pdispa": {"name": "pdispa", "group": "Ungrouped variables", "definition": "['-3','-2','- ','','2','3']", "description": "", "templateType": "anything"}, "pchoicedispa": {"name": "pchoicedispa", "group": "Ungrouped variables", "definition": "pdispa[pchoosea]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pa", "pb", "pc", "pdefs", "ea", "eb", "ec", "ek", "edefs", "ha", "hc", "hdefs", "ldefs", "sign", "choosesign", "signb", "choosesignb", "pchoosea", "pachoices", "hchoosea", "pdispa", "pchoicedispa"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',hdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',edefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/pdapbxee',ldefs)}

"], "matrix": [0, "1", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "exponential equation", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown an exponential function formula and asked to identify the correct graph, given an exponential, a hyperbola, a parabola and a line.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of the following graphs represents $y = \\var{ek} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign}$ ?

", "advice": "

The equation is in the form $y = k \\times a^x + c$, which is an exponential.

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',hdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',edefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/pdapbxee',ldefs)}

"], "matrix": ["0", "0", "1", 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "hyperbola equation", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given the equation of a hyperbola and asked to identify the hyperbolic graph out of a set of 4 graphs.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of the following graphs represents $y = \\frac{\\var{ha}}{x} \\var{choosesign}$?

", "advice": "

This equation represents an inverse relationship, which is graphed as a hyperbola.

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',hdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',edefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/pdapbxee',ldefs)}

"], "matrix": ["1", "0", 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Identify an inverse relationship formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

This is a very simple question with no randomisation.

Students are asked to identify an inverse relationship equation.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The time, measured in hours, for which milk keeps before spoiling ($t$) varies inversely with temperature ($C$), measured in degrees celsius.

", "advice": "

An inverse relationship means that as the input variable increases, the output variable decreases. This means that the input variable needs to be on the denominator of a fraction.

In this question, temperature ($C$) is the input variable, so the equation must be of the form $t = \\frac{k}{C}$.

", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

which of the following equations correctly represents the relationship between $t$ and $C$?

$k$ is a constant.

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Students are shown 4 exponential equations, and 4 graphs and asked to match them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Match the equations with the graphs

", "advice": "

Exponential functions with a power of $x$ exhibit exponential growth - that is, $y$ increases as $x$ increases.

Exponential functions with a power of $-x$ exhibit exponential decay - that is, $y$ decreases as $x$ increases.

When $a \\leq 2$ the function grows or decays much more slowly than when $a \\geq 4$

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{geogebra_applet('https://www.geogebra.org/m/peeue5xr',defs1)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',defs2)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',defs3)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',defs4)}

"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the hyperbola", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given 4 equations for hyperbolas and 4 graphs and asked to match them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Match the equations with the graphs

", "advice": "

A minus sign in the formula flips the hyperbola from the top-right/bottom-left quadrants into the top-left/bottom-right quadrants.

A larger value of $a$ means that the hyperbola's curve is less sharp.

Adding a constant ($+c$ or $-c$) to the equation moves the horizontal asypmtote up or down.

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{geogebra_applet('https://www.geogebra.org/m/rpahpr69',defs1)}

", "

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',defs2)}

", "

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',defs3)}

", "

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',defs4)}

"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the parabola", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given 4 parabola equations and their respective graphs and asked to match them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Match the equations with the graphs

", "advice": "

The general equation for a parabola is $y=ax^w+bx+c$.

The larger the value of $a$, the steeper the parabola.

If $a>0$ then the parabola is concave up (smiling).

If $a<0$ then the parabola is concave down (frowning).

The constant $c$ moves the parabola up or down the $x$-axis by a value of $c$.

If $a>0$, the addition of a $+bx$ term moves the parabola down and to the left, and the addition of a $-bx$ term moves the parabola down and to the right.

If $a<0$, the addition of a $+bx$ term moves the parabola up and to the right, and the addition of a $-bx$ term moves the parabola up and to the left.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "pdefs1": {"name": "pdefs1", "group": "Ungrouped variables", "definition": "[\n ['a',a1],['b',0],['c',0]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdefs2": {"name": "pdefs2", "group": "Ungrouped variables", "definition": "[\n ['a',a2],['b',0],['c',c2]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdefs3": {"name": "pdefs3", "group": "Ungrouped variables", "definition": "[\n ['a',a3],['b',b3],['c',0]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdefs4": {"name": "pdefs4", "group": "Ungrouped variables", "definition": "[\n ['a',a4],['b',b4],['c',c4]\n ]", "description": "", "templateType": "anything", "can_override": false}, "signc2": {"name": "signc2", "group": "Ungrouped variables", "definition": "['{c2}','','+ {c2}']", "description": "", "templateType": "anything", "can_override": false}, "signb3": {"name": "signb3", "group": "Ungrouped variables", "definition": "['-3','-2','-','','+','+2','+3']", "description": "", "templateType": "anything", "can_override": false}, "signb4": {"name": "signb4", "group": "Ungrouped variables", "definition": "['-3','-2','-','','+','+2','+3']", "description": "", "templateType": "anything", "can_override": false}, "signc4": {"name": "signc4", "group": "Ungrouped variables", "definition": "['{c4}','','+ {c4}']", "description": "", "templateType": "anything", "can_override": false}, "dispc2": {"name": "dispc2", "group": "Ungrouped variables", "definition": "signc2[sgn(c2)+1]", "description": "", "templateType": "anything", "can_override": false}, "dispb3": {"name": "dispb3", "group": "Ungrouped variables", "definition": "signb3[b3+3]", "description": "", "templateType": "anything", "can_override": false}, "dispb4": {"name": "dispb4", "group": "Ungrouped variables", "definition": "signb4[b4+3]", "description": "", "templateType": "anything", "can_override": false}, "dispc4": {"name": "dispc4", "group": "Ungrouped variables", "definition": "signc4[sgn(c4)+1]", "description": "", "templateType": "anything", "can_override": false}, "signa1": {"name": "signa1", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "dispa1": {"name": "dispa1", "group": "Ungrouped variables", "definition": "signa1[sgn(a1)+1]", "description": "", "templateType": "anything", "can_override": false}, "signa2": {"name": "signa2", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "signa3": {"name": "signa3", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "signa4": {"name": "signa4", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "dispa2": {"name": "dispa2", "group": "Ungrouped variables", "definition": "signa2[sgn(a2)+1]", "description": "", "templateType": "anything", "can_override": false}, "dispa3": {"name": "dispa3", "group": "Ungrouped variables", "definition": "signa3[sgn(a3)+1]", "description": "", "templateType": "anything", "can_override": false}, "dispa4": {"name": "dispa4", "group": "Ungrouped variables", "definition": "signa4[sgn(a4)+1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b3", "b4", "c2", "c4", "pdefs1", "pdefs2", "pdefs3", "pdefs4", "signc2", "signb3", "signb4", "signc4", "dispc2", "dispb3", "dispb4", "dispc4", "signa1", "dispa1", "signa2", "signa3", "signa4", "dispa2", "dispa3", "dispa4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$y = \\var{dispa1} x^2$", "$y = \\var{dispa2} x^2 \\var{dispc2}$", "$y = \\var{dispa3} x^2 \\var{dispb3} x$", "$y = \\var{dispa4} x^2 \\var{dispb4} x \\var{dispc4}$"], "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs1)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs2)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs3)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs4)}

"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "pick the exponential", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown an exponential graph and asked to identify the equation from a selection of 4.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of these formuale might be represented by this graph?

{geogebra_applet('https://www.geogebra.org/m/a2bvpj8m',edefs)}

", "advice": "

The graph is an exponential function.

If it is showing exponential growth, then the index in the equation is $x$

If it is showing exponential decay, then the index in the equation is $-x$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pachoices": {"name": "pachoices", "group": "Ungrouped variables", "definition": "[-3,-2,-1,1,2,3]", "description": "", "templateType": "anything", "can_override": false}, "pchoosea": {"name": "pchoosea", "group": "Ungrouped variables", "definition": "random(0..5)", "description": "", "templateType": "anything", "can_override": false}, "pa": {"name": "pa", "group": "Ungrouped variables", "definition": "pachoices[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "edefs": {"name": "edefs", "group": "Ungrouped variables", "definition": "[\n ['a',ea],['b',eb],['c',ec],['k',ek]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdispastring": {"name": "pdispastring", "group": "Ungrouped variables", "definition": "['-3','-2','- ','','2','3']", "description": "", "templateType": "anything", "can_override": false}, "pdispa": {"name": "pdispa", "group": "Ungrouped variables", "definition": "pdispastring[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "hk": {"name": "hk", "group": "Ungrouped variables", "definition": "random(-5,-4,-3,-2,-1,1,2,3,4,5)", "description": "", "templateType": "anything", "can_override": false}, "hc": {"name": "hc", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "pdispaneg": {"name": "pdispaneg", "group": "Ungrouped variables", "definition": "pdispastring[5-pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "ea": {"name": "ea", "group": "Ungrouped variables", "definition": "random(11..100)/10", "description": "", "templateType": "anything", "can_override": false}, "eb": {"name": "eb", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "ec": {"name": "ec", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "ek": {"name": "ek", "group": "Ungrouped variables", "definition": "random(-100..100)/10", "description": "", "templateType": "anything", "can_override": false}, "dispebstring": {"name": "dispebstring", "group": "Ungrouped variables", "definition": "['-','']", "description": "", "templateType": "anything", "can_override": false}, "dispeb": {"name": "dispeb", "group": "Ungrouped variables", "definition": "dispebstring[(eb+1)/2]", "description": "", "templateType": "anything", "can_override": false}, "dispebneg": {"name": "dispebneg", "group": "Ungrouped variables", "definition": "dispebstring[1-(eb+1)/2]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pachoices", "pchoosea", "pa", "pb", "pc", "edefs", "pdispastring", "pdispa", "hc", "pdispaneg", "hk", "ea", "eb", "ec", "ek", "dispebstring", "dispeb", "dispebneg"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y = \\var{pdispa} x^2 + \\var{pb}x + \\var{pc}$", "$y = \\var{ek} \\times \\var{ea}^{\\var{dispebneg} x} $", "$y = \\var{ek} \\times \\var{ea}^{\\var{dispeb} x} $", "$y = \\frac{\\var{hk}}{x} + \\var{hc}$"], "matrix": ["0", 0, "1", "1"], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "pick the hyperbola", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown a hyperbola and asked to identify its equation from 4 choices.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of these formuale might be represented by this graph?

{geogebra_applet('https://www.geogebra.org/m/fbzq6qkz',hdefs)}

", "advice": "

This graph is a hyperbola. To identify its equation you need to

(a) see what quadrants its branches are in. If it has a branch in the top-left, then the value of $k$ is positive. Otherwise, there is a minus sign in front of the value for $k$.

(b) look at where its horizontal asymptote is. This value is the value of $c$ in the equation.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pachoices": {"name": "pachoices", "group": "Ungrouped variables", "definition": "[-3,-2,-1,1,2,3]", "description": "", "templateType": "anything", "can_override": false}, "pchoosea": {"name": "pchoosea", "group": "Ungrouped variables", "definition": "random(0..5)", "description": "", "templateType": "anything", "can_override": false}, "pa": {"name": "pa", "group": "Ungrouped variables", "definition": "pachoices[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "hdefs": {"name": "hdefs", "group": "Ungrouped variables", "definition": "[\n ['a',hk],['c',hc]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdispastring": {"name": "pdispastring", "group": "Ungrouped variables", "definition": "['-3','-2','- ','','2','3']", "description": "", "templateType": "anything", "can_override": false}, "pdispa": {"name": "pdispa", "group": "Ungrouped variables", "definition": "pdispastring[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "hk": {"name": "hk", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "hc": {"name": "hc", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "pdispaneg": {"name": "pdispaneg", "group": "Ungrouped variables", "definition": "pdispastring[5-pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "neghk": {"name": "neghk", "group": "Ungrouped variables", "definition": "hk * (-1)", "description": "", "templateType": "anything", "can_override": false}, "neghc": {"name": "neghc", "group": "Ungrouped variables", "definition": "hc * (-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pachoices", "pchoosea", "pa", "pb", "pc", "hdefs", "pdispastring", "pdispa", "hc", "pdispaneg", "hk", "neghk", "neghc"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y = \\var{pdispa} x^2 + \\var{pb} x+ \\var{pc}$", "$y = \\frac{\\var{neghk}}{x} + \\var{hc}$", "$y = \\frac{\\var{hk}}{x} \\var{neghc}$", "$y = \\frac{\\var{hk}}{x} + \\var{hc}$"], "matrix": ["0", 0, 0, "1"], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "pick the parabola", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown a parabola and asked to identify its graph from a selection of choices.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of these formuale might be represented by this graph?

{geogebra_applet('https://www.geogebra.org/m/wu4fe7gg',pdefs)}

", "advice": "

The graph is a parabola. The sign in front of $a$ tells you whether it is concave up ($a>0$) or convave down($a<0$).

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pachoices": {"name": "pachoices", "group": "Ungrouped variables", "definition": "[-3,-2,-1,1,2,3]", "description": "", "templateType": "anything", "can_override": false}, "pchoosea": {"name": "pchoosea", "group": "Ungrouped variables", "definition": "random(0..5)", "description": "", "templateType": "anything", "can_override": false}, "pa": {"name": "pa", "group": "Ungrouped variables", "definition": "pachoices[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "pdefs": {"name": "pdefs", "group": "Ungrouped variables", "definition": "[\n ['a',pa],['b',pb],['c',pc]\n ]", "description": "", "templateType": "anything", "can_override": false}, "pdispastring": {"name": "pdispastring", "group": "Ungrouped variables", "definition": "['-3','-2','- ','','2','3']", "description": "", "templateType": "anything", "can_override": false}, "pdispa": {"name": "pdispa", "group": "Ungrouped variables", "definition": "pdispastring[pchoosea]", "description": "", "templateType": "anything", "can_override": false}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "v3": {"name": "v3", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "v4": {"name": "v4", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "pdispaneg": {"name": "pdispaneg", "group": "Ungrouped variables", "definition": "pdispastring[5-pchoosea]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pachoices", "pchoosea", "pa", "pb", "pc", "pdefs", "pdispastring", "pdispa", "v1", "v2", "v4", "pdispaneg", "v3"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y = \\var{pdispa} x^2 + \\var{pb}x + \\var{pc}$", "$y = \\var{pdispaneg} x^2 + \\var{pb}x + \\var{pc}$", "$y = \\var{v1} \\times \\var{v2}^x$", "$y = \\frac{\\var{v3}}{x} + \\var{v4}$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the exponential formula", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown an exponential graph and asked to choose the correct formula for the graph.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

{geogebra_applet('https://www.geogebra.org/m/fyfns8hu',defs)}

", "advice": "

The exponential curve is {direction}. So we are looking for a formula that has {pmprefix} before the $x$.

When $x=0$, $y=\\var{ek}$. So we are looking for a formula in which $k=\\var{ek}$, that is, $y = \\var{ek} \\times a^{\\var{chooseprefix} x}$.

Thus, the formula we need is $y = \\var{ek} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ea": {"name": "ea", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "ek": {"name": "ek", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "chooseprefix": {"name": "chooseprefix", "group": "Ungrouped variables", "definition": "prefix[(eb+1)/2]", "description": "", "templateType": "anything", "can_override": false}, "choosesign": {"name": "choosesign", "group": "Ungrouped variables", "definition": "sign[sgn(ec)+1]", "description": "", "templateType": "anything", "can_override": false}, "eb": {"name": "eb", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "ec": {"name": "ec", "group": "Ungrouped variables", "definition": "random(-1..3)", "description": "", "templateType": "anything", "can_override": false}, "prefix": {"name": "prefix", "group": "Ungrouped variables", "definition": "['-','']", "description": "", "templateType": "anything", "can_override": false}, "": {"name": "", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}, "sign": {"name": "sign", "group": "Ungrouped variables", "definition": "['{ec}','','+ {ec}']", "description": "", "templateType": "anything", "can_override": false}, "ek2": {"name": "ek2", "group": "Ungrouped variables", "definition": "2*ek", "description": "", "templateType": "anything", "can_override": false}, "ek3": {"name": "ek3", "group": "Ungrouped variables", "definition": "ek/2", "description": "", "templateType": "anything", "can_override": false}, "ec2": {"name": "ec2", "group": "Ungrouped variables", "definition": "ec*(-1)", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n ['a',ea],['b',eb],['c',ec],['k',ek],['xm',6],['ym',40]\n ]", "description": "", "templateType": "anything", "can_override": false}, "choosesign2": {"name": "choosesign2", "group": "Ungrouped variables", "definition": "sign2[sgn(ec2)+1]", "description": "", "templateType": "anything", "can_override": false}, "sign2": {"name": "sign2", "group": "Ungrouped variables", "definition": "['{ec2}','','+ {ec2}']", "description": "", "templateType": "anything", "can_override": false}, "directions": {"name": "directions", "group": "Ungrouped variables", "definition": "['decreasing','increasing']", "description": "", "templateType": "anything", "can_override": false}, "direction": {"name": "direction", "group": "Ungrouped variables", "definition": "directions[(eb+1)/2]", "description": "", "templateType": "anything", "can_override": false}, "pmprefixes": {"name": "pmprefixes", "group": "Ungrouped variables", "definition": "['a -','nothing (meaning +)']", "description": "", "templateType": "anything", "can_override": false}, "pmprefix": {"name": "pmprefix", "group": "Ungrouped variables", "definition": "pmprefixes[(eb+1)/2]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ek", "ea", "chooseprefix", "choosesign", "eb", "ec", "prefix", "", "sign", "ek2", "ek3", "ec2", "defs", "choosesign2", "sign2", "directions", "direction", "pmprefixes", "pmprefix"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Which equation describes this exponential graph?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y = \\var{ek} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign}$", "$y = \\var{ek2} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign}$", "$y = \\var{ek3} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign}$", "$y = \\var{ek2} \\times \\var{ea}^{\\var{chooseprefix} x} \\var{choosesign2}$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the parabola 2", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Student is shown a graph with a parabola and asked to identify the correct equation. Multiple choice question.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Identify the equation that matches the graph

", "advice": "

The parabola's axis of symmetry is the $y$-axis. The parabola cuts the $y$-axis at $y = \\var{c1}$, so the value of $c$ in the formula is $\\var{c1}$.

The parabola is concave {concavity}, so the value of $a$ is {posneg}. This means that the correct formula for the parabola is $y = \\var{dispa1} x^2${dispc1}.

Which of these equations matches this graph?

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs1)}

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y = \\var{dispa1} x^2${dispc1}", "$y = \\var{dispa1} x^2${dispc2}", "$y = \\var{dispa2} x^2${dispc1}", "$y = \\var{dispa2} x^2${dispc2}"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the exponential 2", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are shown an exponential graph and asked to identify its equation from 4 choices.

The graph is randomised.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Identify the equation that matches the graph

", "advice": "

This exponential function has an asymptote at $\\var{c}$, so $c=\\var{c}$.

It cuts the $x$-axis at $c + k = \\var{c+k}$, so $k = \\var{c+k} - (\\var{c}) = \\var{k}$.

Finally the curve is {updownval}, which means that the value of the exponent is {posnegval}.

So the correct equation is $y=\\var{k} \\times \\var{a}^{\\var{bsgn}x} \\var{csgn} \\var{c}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"defs1": {"name": "defs1", "group": "Ungrouped variables", "definition": "[\n ['a',a],['b',b],['c',c],['k',k]\n ]", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(11..19)/10", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "possgn": {"name": "possgn", "group": "Ungrouped variables", "definition": "['','','+']", "description": "", "templateType": "anything", "can_override": false}, "bsgn": {"name": "bsgn", "group": "Ungrouped variables", "definition": "negsgn[sgn(b)+1]", "description": "", "templateType": "anything", "can_override": false}, "csgn": {"name": "csgn", "group": "Ungrouped variables", "definition": "possgn[sgn(c)+1]", "description": "", "templateType": "anything", "can_override": false}, "negsgn": {"name": "negsgn", "group": "Ungrouped variables", "definition": "[\"-\",\"\",\"\"]", "description": "", "templateType": "anything", "can_override": false}, "updown": {"name": "updown", "group": "Ungrouped variables", "definition": "['decaying','','growing']", "description": "", "templateType": "anything", "can_override": false}, "updownval": {"name": "updownval", "group": "Ungrouped variables", "definition": "updown[sgn(b)+1]", "description": "", "templateType": "anything", "can_override": false}, "posneg": {"name": "posneg", "group": "Ungrouped variables", "definition": "['negative','','positive']", "description": "", "templateType": "anything", "can_override": false}, "posnegval": {"name": "posnegval", "group": "Ungrouped variables", "definition": "posneg[sgn(b)+1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["defs1", "a", "b", "c", "k", "possgn", "bsgn", "csgn", "negsgn", "updown", "updownval", "posneg", "posnegval"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Which of these equations matches this graph?

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',defs1)}

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y=\\var{k} \\times \\var{a}^{\\var{bsgn}x} \\var{csgn} \\var{c}$", "$y=\\var{k}^{\\var{bsgn}x}\\var{csgn} \\var{c}$", "$y=\\var{k} \\times \\var{a}^{\\var{negsgn[sgn(b)*(-1)+1]}x}\\var{csgn} \\var{c}$", "$y=\\var{k+c} \\times \\var{a}^{\\var{bsgn}x}$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the hyperbola 2", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given a hyperbolic graph and asked to select the correct equation from 4 choices.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Select the correct equation.

", "advice": "

The hyperbola has an asymptote at $\\var{c}$, so $c=\\var{c}$

The hyperbola has a branch in the top {location} so $a$ must be {posneg}.

Therefore the equation of the hyperbola is $y=\\frac{\\var{a}}{x} \\var{csgn} \\var{c}$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5,-4,-3,-2,-1,1,2,3,4,5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "negsgn": {"name": "negsgn", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "defs1": {"name": "defs1", "group": "Ungrouped variables", "definition": "[\n ['a',a],['c',c]\n ]", "description": "", "templateType": "anything", "can_override": false}, "possgn": {"name": "possgn", "group": "Ungrouped variables", "definition": "['','','+']", "description": "", "templateType": "anything", "can_override": false}, "csgn": {"name": "csgn", "group": "Ungrouped variables", "definition": "possgn[sgn(c)+1]", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "c*(-1)", "description": "", "templateType": "anything", "can_override": false}, "c2sgn": {"name": "c2sgn", "group": "Ungrouped variables", "definition": "possgn[sgn(c2)+1]", "description": "", "templateType": "anything", "can_override": false}, "locationoptions": {"name": "locationoptions", "group": "Ungrouped variables", "definition": "['left','','right']", "description": "", "templateType": "anything", "can_override": false}, "location": {"name": "location", "group": "Ungrouped variables", "definition": "locationoptions[sgn(a)+1]", "description": "", "templateType": "anything", "can_override": false}, "posnegoptions": {"name": "posnegoptions", "group": "Ungrouped variables", "definition": "['negative','','positive']", "description": "", "templateType": "anything", "can_override": false}, "posneg": {"name": "posneg", "group": "Ungrouped variables", "definition": "posnegoptions[sgn(a)+1]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "negsgn", "defs1", "possgn", "csgn", "c2", "c2sgn", "locationoptions", "location", "posnegoptions", "posneg"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Select the equation that represents this graph.

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',defs1)}

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$y=\\frac{\\var{a}}{x} \\var{csgn} \\var{c}$", "$y=\\frac{\\var{a*(-1)}}{x} \\var{csgn} \\var{c}$", "$y=\\frac{\\var{a}}{x} \\var{c2sgn} \\var{c2}$", "$y=\\frac{\\var{a*(-1)}}{x} \\var{c2sgn} \\var{c2}$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "identify the nonlinear graph", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given equations for a line, a parabola, a hyperbola and an exponential, and their respective graphs and are asked to match them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Match the equations with the graphs

", "advice": "

In order to answer this question you need to have learnt the shapes and equations for these four types of relationships.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pa": {"name": "pa", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "pb": {"name": "pb", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "pc": {"name": "pc", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "pdefs": {"name": "pdefs", "group": "Ungrouped variables", "definition": "[\n ['a',pa],['b',pb],['c',pc]\n ]", "description": "", "templateType": "anything", "can_override": false}, "psignb": {"name": "psignb", "group": "Ungrouped variables", "definition": "['-3','-2','-','','+','+2','+3']", "description": "", "templateType": "anything", "can_override": false}, "psignc": {"name": "psignc", "group": "Ungrouped variables", "definition": "['{pc}','','+ {pc}']", "description": "", "templateType": "anything", "can_override": false}, "pdispb": {"name": "pdispb", "group": "Ungrouped variables", "definition": "psignb[pb+3]", "description": "", "templateType": "anything", "can_override": false}, "pdispc": {"name": "pdispc", "group": "Ungrouped variables", "definition": "psignc[sgn(pc)+1]", "description": "", "templateType": "anything", "can_override": false}, "psigna": {"name": "psigna", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "pdispa": {"name": "pdispa", "group": "Ungrouped variables", "definition": "psigna[sgn(pa)+1]", "description": "", "templateType": "anything", "can_override": false}, "ea": {"name": "ea", "group": "Ungrouped variables", "definition": "random(11..60)/10", "description": "", "templateType": "anything", "can_override": false}, "edefs": {"name": "edefs", "group": "Ungrouped variables", "definition": "[\n ['a',ea],['b',eb],['c',0],['k',1]\n ]", "description": "", "templateType": "anything", "can_override": false}, "ha": {"name": "ha", "group": "Ungrouped variables", "definition": "random(-5,-3,3,5)", "description": "", "templateType": "anything", "can_override": false}, "hc": {"name": "hc", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "hcstrings": {"name": "hcstrings", "group": "Ungrouped variables", "definition": "['-3','-2','-1','','+1','+2','+3']", "description": "", "templateType": "anything", "can_override": false}, "hdispc": {"name": "hdispc", "group": "Ungrouped variables", "definition": "hcstrings[hc+3]", "description": "", "templateType": "anything", "can_override": false}, "hdefs": {"name": "hdefs", "group": "Ungrouped variables", "definition": "[\n ['a',ha],['c',hc]\n ]", "description": "", "templateType": "anything", "can_override": false}, "eb": {"name": "eb", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "dispebstring": {"name": "dispebstring", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "diseb": {"name": "diseb", "group": "Ungrouped variables", "definition": "dispebstring[eb+1]", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-3,-2,-1,1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "lb": {"name": "lb", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}, "edispbstring": {"name": "edispbstring", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything", "can_override": false}, "edispb": {"name": "edispb", "group": "Ungrouped variables", "definition": "edispbstring[eb+1]", "description": "", "templateType": "anything", "can_override": false}, "ldispbstring": {"name": "ldispbstring", "group": "Ungrouped variables", "definition": "['-3','-2','-1','0','+1','+2','+3']", "description": "", "templateType": "anything", "can_override": false}, "ldispb": {"name": "ldispb", "group": "Ungrouped variables", "definition": "ldispbstring[lb+3]", "description": "", "templateType": "anything", "can_override": false}, "ldefs": {"name": "ldefs", "group": "Ungrouped variables", "definition": "[\n ['a',m],['b',lb]\n ]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pa", "pb", "pc", "pdefs", "psignb", "psignc", "pdispb", "pdispc", "psigna", "pdispa", "ea", "edefs", "ha", "hc", "hcstrings", "hdispc", "hdefs", "eb", "dispebstring", "diseb", "m", "lb", "edispbstring", "edispb", "ldispbstring", "ldispb", "ldefs"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$y = \\var{m} x \\var{ldispb}$", "$y = \\var{ea} ^{\\var{edispb} x}$", "$y = \\frac{\\var{ha}}{x} \\var{hdispc}$", "$y = \\var{pdispa} x^2 \\var{pdispb} x \\var{pdispc}$"], "matrix": [["1", 0, 0, 0], [0, "1", 0, 0], [0, 0, "1", 0], [0, 0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["

{geogebra_applet('https://www.geogebra.org/m/pdapbxee',ldefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/peeue5xr',edefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/rpahpr69',hdefs)}

", "

{geogebra_applet('https://www.geogebra.org/m/vscpgws8',pdefs)}

"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Stopping distance of a car", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given a formula and asked to evaluate it for a given input value, which is randomised.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The stopping distance of a car, $d$, in metres, can be estimated using the formula $d = \\frac{v^2}{11}$, where $v$ is the speed of the car in metres per second.

", "advice": "

The speed of the car is given by $\\frac{\\var{v} \\times \\var{v}}{11} = \\var{round(v*v/11)}$ m

", "rulesets": {}, "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(11..28)", "description": "

speed in m/s

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What is the stopping distance in metres, if the car is travelling at {v} m/s?

Give your answer rounded to the nearest metre.

", "minValue": "round({v}*{v}/11)", "maxValue": "round({v}*{v}/11)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Identify the y-intercept of an exponential function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given an exponential equation and asked to identify the y-intercept from a list of choices.

The constants in the exponential equation have been randomised.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

What is the $y$-intercept of the exponential function $y=\\var{k} \\times \\var{a}^{\\var{bdisplay}x}$?

", "advice": "

Every exponential of the form $y=a^x$ or $y=a^{-x}$ passes through the point $(0,1)$.

When the equation is multiplied by a constant, $k$, the $y$-intercept moves to pass through the point $(0,k)$.

So this exponential has a $y$-intercept of $(0,\\var{k})$

exponent base

", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(21 .. 50)/10", "description": "", "templateType": "anything", "can_override": false}, "bdspchoices": {"name": "bdspchoices", "group": "Ungrouped variables", "definition": "['-','','']", "description": "

if b is -1 then display an index of -x. Otherwise just display x.

", "templateType": "anything", "can_override": false}, "bdisplay": {"name": "bdisplay", "group": "Ungrouped variables", "definition": "bdspchoices[sgn(b)+1]", "description": "

if b is -1 then display an index of -x. Otherwise just display x.

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "k", "bdspchoices", "bdisplay"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$(0,\\var{k})$", "$(0,\\var{a})$", "$(0,1)$", "$(0,0)$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Draw an exponential", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are asked to draw an exponential from an equation by drawing a table of graphs and plotting them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

This question asks you to draw a graph. You need to mark your own work for this question after you have done it.

", "advice": "

$y=\\var{a}^x \\var{cdisp}$

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

$x$	0	1	2	3	4	5
$y$	{k+c}	{k*a+c}	{k*a^2+c}	{k*a^3+c}	{k*a^4+c}	{k*a^5+c}

{geogebra_applet('https://www.geogebra.org/m/fyfns8hu',defs)}

", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(11..19)/10", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-1..3)", "description": "", "templateType": "anything"}, "cdispoptions": {"name": "cdispoptions", "group": "Ungrouped variables", "definition": "['-3','-2','-1','','+1','+2','+3']", "description": "", "templateType": "anything"}, "cdisp": {"name": "cdisp", "group": "Ungrouped variables", "definition": "cdispoptions[c+3]", "description": "", "templateType": "anything"}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n ['k',k],['a',a],['b',1],['c',c],['xm',6],['ym',maxval * 1.1]\n ]", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything"}, "negdisp": {"name": "negdisp", "group": "Ungrouped variables", "definition": "['-','','']", "description": "", "templateType": "anything"}, "bdisp": {"name": "bdisp", "group": "Ungrouped variables", "definition": "negdisp[sgn(b)+1]", "description": "", "templateType": "anything"}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "maxval": {"name": "maxval", "group": "Ungrouped variables", "definition": "k*a^5+c", "description": "

The maximum value to be plotted

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "cdispoptions", "cdisp", "defs", "b", "negdisp", "bdisp", "k", "maxval"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Generate a table of values and use these to plot a graph of $y=\\var{a}^x \\var{cdisp}$ between $x=0$ and $x=5$.

Click on the \"Reveal Answers\" button to see a solution

"}], "partsMode": "explore", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Draw a inverse relationship", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are asked to draw a hyperbola from an equation by drawing a table of graphs and plotting them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

This question asks you to draw a graph. You need to mark your own work for this question after you have done it.

", "advice": "

$y=\\frac{\\var{a}}{x}$

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

$x$

$y$

undefined:

$y=0$ is an asymptote

{a}

{a/2}

{a/3}

{a/4}

{a/5}

{geogebra_applet('https://www.geogebra.org/m/w8tssh2t',defs)}

", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1,2,3,4,5)", "description": "", "templateType": "anything"}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n ['a',a], ['c',0],['xmin',-1],['ymin',-1],['xmax',6],['ymax',2*a]\n ]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "defs"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Generate a table of values and use these to plot a graph of $y=\\frac{\\var{a}}{x}$ between $x=0$ and $x=5$.

Click on the \"Reveal Answers\" button to see a solution

"}], "partsMode": "explore", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Draw a parabola", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are asked to draw a parabola from an equation by drawing a table of graphs and plotting them.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

This question asks you to draw a graph. You need to mark your own work for this question after you have done it.

", "advice": "

$y=\\var{a}x^2 \\var{cdisp}$

\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\n\n\n

$x$	-10	-8	-6	-4	-2	0	2	4	6	8	10
$y$	{a*100+c}	{a*64+c}	{a*36+c}	{a*16+c}	{a*4+c}	{c}	{a*4+c}	{a*16+c}	{a*36+c}	{a*64+c}	{a*100+c}

{geogebra_applet('https://www.geogebra.org/m/wu4fe7gg',defs)}

", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-0.3,-0.2,-0.1,0.1,0.2,0.3)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything"}, "cdispoptions": {"name": "cdispoptions", "group": "Ungrouped variables", "definition": "['-3','-2','-1','','+1','+2','+3']", "description": "", "templateType": "anything"}, "cdisp": {"name": "cdisp", "group": "Ungrouped variables", "definition": "cdispoptions[c+3]", "description": "", "templateType": "anything"}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n ['a',a],['b',0],['c',c]\n ]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "cdispoptions", "cdisp", "defs"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Generate a table of values and use these to plot a graph of $y=\\var{a}x^2 \\var{cdisp}$ between $x=-10$ and $x=10$.

Click on the \"Reveal Answers\" button to see a solution

"}], "partsMode": "explore", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Identify the point lying on a parabola", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given a parabola equation and 4 points. They need to determine which point lies on the parabola.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Which of the following points lies on the parabola $y = \\var{a} x^2 - \\var{b}x - \\var{c}$?

", "advice": "

The best way to approach this is to substitute the points into the equation and see which one works.

You will observe that in all four choices, $x = \\var{x}$.

If we substitute $x=\\var{x}$ into the equation $y = \\var{a} x^2 - \\var{b} x -\\var{c}$ we find that $y = \\var{a*x*x-b*x-c})$.

So the point that lieson the parabola is $(\\var{x},\\var{a*x*x-b*x-c})$.

parabola constant

", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "

parabola constant

", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "

parabola constant

", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-2,-1,0,1,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "x"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$(\\var{x},\\var{(a*x*x)-(b*x)-c})$", "$(\\var{x},\\var{a*x*x+b*x+c})$", "$(\\var{x},\\var{-a*x*x+b*x+c})$", "$(\\var{x},\\var{-a*x*x-b*x-c})$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Group", "pickingStrategy": "random-subset", "pickQuestions": "2", "questionNames": ["", "", "", "", "", "", "", ""], "questions": [{"name": "Exponential population growth from formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

Students are given an exponential equation and asked to evaluate it at two points.

The constants in the exponential are randomised.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

A population of {objects} has a population, $P$, that can be described by the function $P = \\var{k} \\times \\var{a}^t $, where $t$ is the time in {timeperiod}.

", "advice": "

When $t=0$,

$ P = \\var{k} \\times \\var{a}^t = \\var{k} \\times \\var{a}^0 = 1$

When $t = \\var{time}$,

$P = \\var{k} \\times \\var{a}^\\var{time} = \\var{answer_raw}$

When this is rounded to the nearest whole number, we get

$P = \\var{answer}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"objectlist": {"name": "objectlist", "group": "Ungrouped variables", "definition": "['bacteria','ants','moths','viruses','mould spores']", "description": "", "templateType": "anything", "can_override": false}, "objectchoice": {"name": "objectchoice", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "object": {"name": "object", "group": "Ungrouped variables", "definition": "objectlist[objectchoice]", "description": "", "templateType": "anything", "can_override": false}, "timeperiodlist": {"name": "timeperiodlist", "group": "Ungrouped variables", "definition": "['days','weeks','months','years']", "description": "", "templateType": "anything", "can_override": false}, "timeperiodchoice": {"name": "timeperiodchoice", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "timeperiod": {"name": "timeperiod", "group": "Ungrouped variables", "definition": "timeperiodlist[timeperiodchoice]", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..10000)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(11..50)/10", "description": "", "templateType": "anything", "can_override": false}, "time": {"name": "time", "group": "Ungrouped variables", "definition": "random(2..15)", "description": "", "templateType": "anything", "can_override": false}, "answer_raw": {"name": "answer_raw", "group": "Ungrouped variables", "definition": "k*a^time", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "round(answer_raw)", "description": "", "templateType": "anything", "can_override": false}, "objects": {"name": "objects", "group": "Ungrouped variables", "definition": "", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["objectlist", "objectchoice", "object", "timeperiodlist", "timeperiodchoice", "timeperiod", "k", "a", "time", "answer_raw", "answer", "objects"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What is the initial population, $P$, when $t$ = 0?

", "minValue": "{k}", "maxValue": "{k}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What is the population, $P$, when $t = \\var{time}$ {timeperiod}

Round your answer to the nearest whole number.

", "minValue": "round({k}*{a}^{time})", "maxValue": "round({k}*{a}^{time})", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Exponential population growth from graph", "extensions": ["geogebra"], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "

The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.

", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates to true or false.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "

The student is shown an exponential graph and asked to evaluate the function at some given value.

They are also asked whether or not the model is valid for all real inputs, but they only give a yes/no response. The reasoning is explained in the advice but is not required from students.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The following graph shows the growth of a population, $P$ of {object} as a function of time, $t$, given in {timeperiod}.

", "advice": "

The value of $P$ when $t=\\var{time}$ can be read off the graph.

We go along the horizontal axis until we find $\\var{time}$.

Then we go vertically up until we meet the graph.

Then we go left until we reach the vertical axis, and we read off the value there, which is approximately $\\var{answer1}$.

This graph is not valid as a model for all values of $t$.

Firstly, it does not normally make sense to talk about negative values of time, so the graph is only valid for $t \\geq 0$.

Secondly, according to this graph, the population would keep increasing exponentially forever. This cannot be the case. At some point the population will slow down and/or stop its growth rate, and this model will no longer apply.

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{geogebra_applet('https://www.geogebra.org/m/fyfns8hu',defs)}

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What is the value of the population $P$ after $\\var{time}$ {timeperiod}?

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Is this model valid for all values of $t$? Why or why not?

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Shows an exponential graph displaying the value of an investment under compound interest. Students have to identify the type of graph and answer 3 questions about the value of the investment from the graph.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The following graph shows the value of an investment that is compounded at {rates[idx]} % per annum.

The x-axis shows the time of investment in years, and the y-axis shows the investment value.

{image('resources/question-resources/'+images[idx])}

", "advice": "

This is an exponential graph.

The initial value of the investment is the value where the graph cuts the $y$-axis. This is the value of the investment when time = 0, at the start of the investment. This is \\${initials[idx]}.

The investment starts at \\${initials[idx]} so to double it needs to reach \\${2*initials[idx]}.

We find \\${2*initials[idx]} on the $y$-axis. Then we go across until we meet the graph. Then we go down to the $x$ axis and read off the value there. We need to be careful doing this, because the marker lines on the axis may not be 1 year apart. In this case, the marker lines are {gradations[idx]} apart, and the time for the investment to double is {doubles[idx]} years.

Observe that {2*investment} = 2 $\\times$ {investment}.

Because this is an exponential graph, the time taken for the investment to double is constant. Therefore the time taken for an initial investment of \\${investment} to reach \\${2*investment} is {doubles[idx]} years.

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What type of graph is this?

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What is the initial value of the investment in dollars?

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How many years does it take for the value of the investment to double?

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If \\${investment} was invested at this same interest rate, how many years would it take for the value of the investment to reach \\${2*investment}?

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Data from a compound interest are shown in a table. Students are asked to compute the value of an investment, and to identify the type of graph.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

A principal amount of \\${principal} was invested for {periods} {timeperiods[timeidx]}s at an interest rate of {rate}% per annum.

The following table shows the amount that \\$1000 invested under the same scheme would be worth at the end of each {timeperiods[timeidx]}.

\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\n\n\n

{timeperiods[timeidx]}s	0	1	2	3	4	5	6	7	8	9	10
investment value ($)	\\$1000	{values[1]}	{values[2]}	{values[3]}	{values[4]}	{values[5]}	{values[6]}	{values[7]}	{values[8]}	{values[9]}	{values[10]}

", "advice": "

The value of \\$1000 invested for {periods} {timeperiods[timeidx]}s at an interest rate of {rate}% per annum is \\${round((1000 * (1 + eff_rate/100)^periods)*100)/100}.

The principal actually invested was {principal / 1000} times this investment, so we need to multiply this value by {principal / 1000}.

{principal / 1000} $\\times$ \\${round((1000 (1 + eff_rate/100)^periods)*100)/100} = \\${(principal / 1000) * round(1000*((1 + eff_rate/100)^periods)*100)/100}.

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What would the \\${principal} investment be worth at the end of {periods} {timeperiods[timeidx]}s?

Give your answer to the nearest cent. Do not enter the dollar sign.

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If these data were plotted, what type of graph would they lie on?

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Students are given a formula for compound interest and asked to evaluate it for given initial investment values.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

A formula for compound interest is given by: $V = I \\times (1+A)^t$,

where

$V$ is the value of the investment after $t$ years,

$A$ is the interest rate per annum,

$I$ is the amount invested.

", "advice": "

We need to calculate $ \\var{principal} \\times (1 + \\var{rate})^\\var{years} = $ {answer}

This is an exponentially increasing function.

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Compute the value of an investment after {years} years if \\${principal} is invested at an interest rate of {rate}.

Give your answer in dollars and cents. Only enter the number; do not include the dollar sign in your answer.

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Which type of curve best represents the value $V$ of this investment over time?

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A bill (for a random amount) is divided equally between a number of people.

Students evaluate the per-person cost for a random number of people.

They then complete a table showing the cost per person for various numbers of people.

Finally they are asked to identify the shape of the graph that represents this function. They are given 2 exponentials, a hyperbola and a line to choose from.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

A share-house contains {n} people.

They receive a bill for \\${total}.

", "advice": "

Each person pays $\\var{total} \\div \\var{n} = \\var{total/n}$ When we round this to the nearest cent, each person pays {totalmoney}.

As the number of people increases, the cost per person decreases.

This is an inverse relationship which is represented by the positive half of a hyperbola. It has asymptotes at $x=0$ and $y=0$ so it never crosses the $y$-axis (unlike the decaying exponential function).

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If each person in the house pays an equal share of the bill, how much does each person pay?

Give your answer in dollars, rounded to the nearest cent.

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Now let's explore what happens if there is a different number of people in the share-house.

Complete this table showing how much each person has to pay in each case.

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

number of people	1	2	3	4	5
amount to pay	{total}	[[0]]	[[1]]	[[2]]	[[3]]

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Which of the following graphs could represent this relationship?

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", "

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Students need to substitute a value into an equation and solve it. The equation constant (gravity) and the value (time) are both randomised.

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On {planets[idx]}, the equation

$d=\\var{gravities[idx]/2}t^2$

can be used to express the distance ($d$ metres) that an object falls in $t$ seconds, if air resistance is ignored.

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The distance fallen is $d=\\var{gravities[idx]/2} \\var{t}^2 = \\var{d} $ metres, which is rounded off to $\\var{round(d)}$ metres.

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fall time in seconds

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Fall distance

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How far does an object fall in {t} seconds on {planets[idx]}?

Round your answer to the nearest whole number of metres.

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Students explore the relationship between length and area of a rectangle.

The perimeter of the rectangle is randomised. Students are given 11 different lengths, and asked to compute rectangle width and area for each. They are then asked to graph the function, identify it as a parabola, and estimate the maximum value.

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You have $\\var{p}$ metres of fencing with which to build a rectangular pen.

You are going to explore the amount of area within the pen that you can achieve with different side lengths.

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$Perimeter = 2 \\times length + 2 \\times width$.

So $width = (perimeter - 2 \\times length)\\div 2$

$Area = length \\times width$

The graph of area plotted against length looks like this:

This is a parabola.

The maximum value from the table is {maxareafromtable}. The maximum area from the graph occurs when the length is {xmax} and is equal to {ymax}.

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the perimeter

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first length

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the shortest length

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Complete the following table.

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

length	{lengths[0]}	{lengths[1]}	{lengths[2]}	{lengths[3]}	{lengths[4]}	{lengths[5]}	{lengths[6]}	{lengths[7]}	{lengths[8]}	{lengths[9]}	{lengths[10]}
width	[[0]]	[[1]]	[[2]]	[[3]]	[[4]]	[[5]]	[[6]]	[[7]]	[[8]]	[[9]]	[[10]]
area	[[11]]	[[12]]	[[13]]	[[14]]	[[15]]	[[16]]	[[17]]	[[18]]	[[19]]	[[20]]	[[21]]

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Now use the information in the table to plot a graph of area against length.

At the end, click the \"Reveal answers\" button to see a sample graph.

What form does the graph have?

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Estimate the maximum area from your table or graph.

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This set of 10 practice questions will change each time that you open them.

After you have attempted a question, press \"submit answer\" and it will tell you whether or not you are correct.

After you have attempted a question, you can see a worked solution. Press \"reveal answers\", then click \"OK\" on the popup message. A worked solution will appear at the bottom of the screen.

If you would like more practice on a particular type of question, click \"Try another question like this one\", and click \"OK\" on the popup message. A new version of the same question will appear.

If your screen is large enough, you can go to any question that you wish via the menu on the left hand side. Otherwise, you can scroll through the questions using the arrow buttons at the top left of the screen.

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The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.

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