// Numbas version: exam_results_page_options
{"name": "Quadratic functions: Working with the Formula - Fitness of Population", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Quadratic functions: Working with the Formula - Fitness of Population", "tags": [], "metadata": {"description": "<p>The relationship between the frequency of an allele A, $x$, at a genetic locus in a diploid population and the fitness of a population with this frequency of allele A, $w$, is described by the function $w=ax^2+x(b-x)+c(b-x)^2$ . The aims are (a) ti simplify the algebraic expression, (b) calculate the fitness of a population with a given allele A frequency, and (c) calculate the allele A frequency when the fitness of the population is given.</p>", "licence": "None specified"}, "statement": "<p>Let $x$ represent the frequency of an allele A at a genetic locus in a diploid population. The fitness, $w$, of a population with this frequency of allele A is given by:</p>\n<p>\\[ \\simplify{w={a}x^2+x({b}-x)+{c}({b}-x)^2} \\]</p>", "advice": "<p>a) We need to open the brackets and collect like terms:</p>\n<p>\\[ \\begin{split} w &amp;= \\simplify{{a}x^2+x({b}-x)+{c}({b}-x)^2} \\\\ \\implies w &amp;= \\simplify[!basic]{{a}x^2+x({b}-x)+{c}({b}-x)({b}-x)}\\\\ \\implies w &amp;= \\simplify [unitFactor, !basic] {{a}x^2+x({b}-x)+{c}({b}^2-{b}x-{b}x+x^2)} \\\\ \\implies w &amp;= \\simplify {{a}x^2+x({b}-x)+{c}({b}^2-{b}x-{b}x+x^2)} \\\\ \\implies w &amp;= \\simplify[unitFactor, !collectNumbers] {{a}x^2+{b}*x-x^2+{c}*{b^2}-{c}*{2*b}x+{c}x^2} \\\\ \\implies w &amp;=\\simplify[all, !noLeadingMinus] {{a}x^2+{b}*x-x^2+{c}*{b}^2-2*{c}*{b}x+{c}x^2} \\end{split}\\]</p>\n<p>b) We substitute $x=\\var{x_1}$ in the simplified formula:</p>\n<p>\\[w=\\simplify[unitfactor, !basic]{{coef_a}*{x_1}^2+{coef_b}*{x_1}+{coef_c}} \\\\ \\implies w=\\simplify [unitfactor, !basic]{{coef_a}*{x_1^2}+{coef_b}*{x_1}+{coef_c}} \\\\ \\implies w=\\var{w_ans_rounded} \\text{&nbsp; (2 d.p.)}\\]</p>\n<p>Note: we get the same result regardless of which version of the formula we use. Using the simplified formula however is quicker.</p>\n<p>c) We need to substitute $w=\\var{w_1}$ and solve the quadratic equation:</p>\n<p>\\[ \\simplify[!noLeadingMinus]{{w_1}={coef_a}*x^2+{coef_b}*x+{coef_c}} \\\\ \\implies \\simplify[!noLeadingMinus]{{w_1}-{coef_a}*x^2-{coef_b}*x-{coef_c}=0} \\\\ \\implies \\simplify{{w_1}-{coef_a}*x^2-{coef_b}*x-{coef_c}=0}\\]</p>\n<p>Using the quadratic formula</p>\n<p>\\[ x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\]</p>\n<p>We get the solutions:</p>\n<p>\\[ \\begin{split} x_1 &amp;\\,= \\frac{\\simplify{-{co_b}+sqrt({co_b}^2-4*{co_a}*{co_c})}}{\\simplify{{2*co_a}}} \\qquad \\text{and} \\qquad&nbsp; x_2 &amp;\\,= \\frac{\\simplify{-{co_b}-sqrt({co_b}^2-4*{co_a}*{co_c})}}{\\simplify{{2*co_a}}} \\\\\\\\ &amp;\\,= \\var{ansx_1_rounded},\\quad &amp;\\,=\\var{ansx_2_rounded}. \\end{split}\\]</p>\n<p>Taking into consideration the context of the problem, we need to reject one of the two answers if it is negative. Thus, the final answer is $x=\\var{ans_c}$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "problem posing variables", "definition": "random (0 .. 0.5 # 0.1 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "problem posing variables", "definition": "random([1,2])", "description": "", "templateType": "anything", "can_override": false}, "x_1": {"name": "x_1", "group": "problem posing variables", "definition": "random (0..1 #0.05 except [0,1]) ", "description": "", "templateType": "anything", "can_override": false}, "w_1": {"name": "w_1", "group": "problem posing variables", "definition": "random (0..1 #0.1 except [0,1]) ", "description": "", "templateType": "anything", "can_override": false}, "coef_a": {"name": "coef_a", "group": "answer variables", "definition": "{a}+{c}-1", "description": "", "templateType": 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"ansx_1_rounded", "ansx_2_rounded", "ans_c"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "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": "<p>Write $w$ in the form $ax^2+bx+c$.</p>\n<p>$w=$[[0]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, 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