This equation, or its graph, can be described as a
", "minMarks": 0, "shuffleChoices": true, "type": "1_n_2", "maxMarks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "distractors": ["", "", "", "", "", ""], "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "variableReplacements": [], "prompt": "An equation of the form $y=ax^2+bx+c$ is a quadratic and the graph of it is called a parabola.
", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "displayType": "radiogroup", "displayColumns": 0, "scripts": {}, "stepsPenalty": "1", "choices": ["straight line
", "parabola/quadratic
", "cubic
", "hyperbola
", "circle
", "quartic
"], "matrix": [0, "1", 0, 0, 0, 0]}, {"marks": 0, "prompt": "The graph of this equation
", "minMarks": 0, "shuffleChoices": false, "type": "1_n_2", "maxMarks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "variableReplacements": [], "prompt": "The number in front of $x^2$ is called the coefficient of $x^2$ or the leading coefficient because it belongs to the term with the highest power.
\nIf this coefficient is postive then the parabola will be concave up, or 'open upwards', or in other words, look like a smiley face!
\nIf this coefficient is negative then the parabola will be concave down, or 'open downwards', or in other words, look like a sad face!
", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "displayType": "radiogroup", "displayColumns": 0, "scripts": {}, "stepsPenalty": "1", "choices": ["opens upwards (is concave up)
", "opens downwards (is concave down)
"], "matrix": "concave"}, {"marks": 0, "prompt": "The $y$-intercept of the graph is $y=$[[0]].
", "gaps": [{"marks": 1, "showCorrectAnswer": true, "maxValue": "{c}", "type": "numberentry", "minValue": "{c}", "showPrecisionHint": false, "correctAnswerFraction": true, "scripts": {}, "variableReplacements": [], "allowFractions": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "variableReplacements": [], "prompt": "The $y$-intercept is the value of $y$ when $x=0$, that is, the value of $y$ where the graph hits the $y$-axis. To find it, substitute $x=0$ into our equation:
\n\\[y=\\simplify[unitFactor,basic,fractionNumbers]{{a}0^2+{b}0+{c}}=\\var{c}.\\]
\n", "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": "1", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "prompt": "The axis of symmetry of the graph of this function is $x=$[[0]].
\n
The vertex of the graph of this function is at $\\large($ [[0]], [[1]] $\\large)$.
The axis of symmetry of a parabola is the line along which you could reflect the parabola and it would look exactly the same. For the quadratic $y=ax^2+bx+c$, the equation of the axis of symmetry is \\[x=-\\dfrac{b}{2a}.\\] Using our given equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we get $x=\\var{axis_x}$.
\n\nThe vertex, or the turning point, or the stationary point of the parabola is on this line, so it's $x$ coordinate must be $\\var{axis_x}$ and its $y$ coordinate must be the $y$ value that corresponds to this $x$ value after susbstituting it into the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$. That is \\[y=\\simplify[unitFactor,basic,fractionNumbers]{{a}({axis_x})^2+{b}({axis_x})+{c}}=\\var{axis_y}.\\]
", "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": "2", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "prompt": "The set of $x$-intercepts of the graph would be [[0]].
\nNote: If there are no intercepts, enter set()
\nIf there is only one intercept, say $x=5$, enter set(5)
\nIf there are two intercepts, say $x=-2$ and $x=1.5$, enter set(-2,1.5)
\nIf there are three intercepts, say $x=-2$, $x=1.5$ and $x=5$, enter set(-2,1.5,5)
", "gaps": [{"marks": 1, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "type": "jme", "checkvariablenames": false, "variableReplacements": [], "vsetrange": [0, 1], "vsetrangepoints": 5, "answer": "{xints}", "showCorrectAnswer": true, "checkingtype": "absdiff", "scripts": {}, "showpreview": true, "expectedvariablenames": []}], "type": "gapfill", "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "variableReplacements": [], "prompt": "For the quadratic $y=ax^2+bx+c$, the $x$-intercepts (if they exist) are given by \\[x=\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}.\\]
\nNotice how the first term gives the axis of symmetry and the second term says how far away from the axis of symmetry to go in both directions!
\nFor the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we have $b^2-4ac=\\simplify[all,fractionNumbers]{{b}^2-4{a}{c}}=\\var{disc}$ and so there are no $x$-intercepts. is one $x$-intercept: are two $x$-intercepts:
\n| $x$ | \n$=$ | \n$\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}$ | \n
| \n | \n | \n |
| \n | $=$ | \n$\\simplify[basic,unitFactor,fractionNumbers]{{axis_x}}\\pm\\simplify[basic,unitFactor,fractionNumbers]{sqrt{{disc}}/({2*a})}$ | \n
| \n | \n | \n |
| \n | $=$ | \n$\\var{axis_x}$ $\\var{xint0}, \\, \\var{xint1}$ | \n
Given the degree of a polynomial is $2$, the maximum number of possible 'bends' or 'turns' in the graph is [[0]].
", "gaps": [{"marks": 1, "showCorrectAnswer": true, "maxValue": "1", "type": "numberentry", "minValue": "1", "showPrecisionHint": false, "correctAnswerFraction": false, "scripts": {}, "variableReplacements": [], "allowFractions": false, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "steps": [{"marks": 0, "showCorrectAnswer": true, "type": "information", "scripts": {}, "variableReplacements": [], "prompt": "A degree $n$ polynomial has at most $n-1$ bends in its graph.
", "variableReplacementStrategy": "originalfirst"}], "stepsPenalty": "1", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "variable_groups": [], "statement": "You are given the equation $y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$.
", "advice": "", "showQuestionGroupNames": false, "functions": {}, "extensions": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "contributors": [{"name": "Ethan Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/739/"}]}]}], "contributors": [{"name": "Ethan Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/739/"}]}