// Numbas version: exam_results_page_options {"name": "Quadratic Functions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratic Functions 1", "tags": [], "metadata": {"description": "Finding parameters of a factorised quadratic based on x-intercepts at vertex. Function has one intercept at origin and one at a positive integer value. Constant multiplier is randomised and requires vertex in order to calculate.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Consider the quadratic function, $f(x) = px (q−x)$, where $p$ and $q$ are positive integers. The graph of $y = f(x)$ passes through the point ({q}, 0).

", "advice": "

In part (a) note that the factorised form of the equation gives the two vertices. Evaluate when the function is zero.

\n

This occurs at the origin and when $x = q$. Therefore $q =$ {q}

\n

In part (b) $p$ can be evaluated by substituting the values of $x$ and $y$ at the vertex into the equation.

\n

{ver_y} $= p \\times${ver_x}$(${q} $-$ {ver_x}$)$

\n

$p = $ {ver_y} $ \\div [${ver_x}$\\times(${q} $-$ {ver_x}$)]$

\n

$p = $ {p}

\n

In part (c) note that the coefficient of $x^2$ is negative and therefore the parabola is concave down.  

\n

The range can be calculated from the y-coordinate of the vertex since this is a maximum point.

\n

$y \\leq$ {ver_y}

", "rulesets": {}, "extensions": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "

p in equation px(q-x)

", "templateType": "randrange"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "

q in equation px(q-x)

", "templateType": "randrange"}, "ver_x": {"name": "ver_x", "group": "Ungrouped variables", "definition": "{q}/2", "description": "

x-coordinate of vertex, based on q

", "templateType": "anything"}, "ver_y": {"name": "ver_y", "group": "Ungrouped variables", "definition": "{p}*{ver_x}*({q}-{ver_x})", "description": "

y coordinate of vertex, based on substitution into the formula.

", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "q", "ver_x", "ver_y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the value of $q$.

", "minValue": "{q}", "maxValue": "{q}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"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": "

The vertex of the function is at ({ver_x},{ver_y}).

"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the value of $p$.

", "minValue": "{p}", "maxValue": "{p}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en", "eu", "plain-eu"], "correctAnswerStyle": "plain"}, {"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 is the range of $f(x)$

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": false, "choices": ["$y \\leq ${ver_y}", "$y > ${ver_y}", "$y\\leq ${ver_x}", "$x \\in [-\\infty,\\infty]$"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Johnathan Gregg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4974/"}]}]}], "contributors": [{"name": "Johnathan Gregg", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4974/"}]}