Determine a quadratic rule
Start with the quadratic form that displays the exact feature supplied, such as roots or a vertex. Use another point to determine the remaining constant, then expand and verify if the question asks for general form
\[y=a(x-p)(x-q),\quad y=a(x-h)^2+k\]
- Match the given feature Choose a form that displays the information supplied in the question \(\displaystyle x\text{-intercepts}=2,5\Rightarrow y=a(x-2)(x-5)\)
- Use another point Substitute a known point into the chosen form to find the remaining constant or scale \(\displaystyle -4=a(3-2)(3-5)\Rightarrow a=2\)
- Expand exactly Expand the factorised or completed-square form carefully when general form is required \(\displaystyle 2(x-2)(x-5)=2x^2-14x+20\)
- Check the given point Substitute a supplied input into the final rule to check that the stated output is obtained \(\displaystyle y(3)=2(3)^2-14(3)+20=-4\)
Worked example
A quadratic has roots -1 and 4 and passes through the point (2, -6). Determine its rule in general form and check the output when x = 0
\[x\text{-intercepts}=-1,4,\quad y(2)=-6\]
- \(\displaystyle y=a(x+1)(x-4)\)Use the factorised form because the two roots are known
- \(\displaystyle -6=a(2+1)(2-4)=a(3)(-2)=-6a\Rightarrow a=1\)Substitute the known point to determine the scale
- \(\displaystyle y=(x+1)(x-4)=x^2-3x-4\)Expand the factors to write the rule in general form
- \(\displaystyle y(0)=0^2-3(0)-4=-4\)Substitution gives the requested output at x = 0
Common mistakes
Fixing the scale from roots alone
✕\(\displaystyle x\text{-intercepts}=2,5\Rightarrow y=(x-2)(x-5)\)✓\(\displaystyle x\text{-intercepts}=2,5\Rightarrow y=a(x-2)(x-5)\)
The roots determine the two factors, not the scale. Another point is needed to find aReversing a root factor
✕\(\displaystyle x\text{-intercepts}=2,5\Rightarrow y=a(x+2)(x+5)\)✓\(\displaystyle x\text{-intercepts}=2,5\Rightarrow y=a(x-2)(x-5)\)
A root r gives a factor x-r, so the factor sign is opposite to the root valueFitting three points with a line
✕\(\displaystyle (0,2),(1,0),(2,0)\Rightarrow y=-2x+2\)✓\(\displaystyle (0,2),(1,0),(2,0)\Rightarrow y=x^2-3x+2\)
Three points determine a quadratic through three independent coefficient equations, not a linear rule through only two coefficientsDifficulty levels
introUse a vertex or roots with one point to determine a quadratic rule in a displayed form
standardDetermine a rule and convert it between factorised, completed-square and general forms
challengeUse three points or exact information that requires a longer coefficient calculation
examChoose a suitable form, fill a missing value and verify the resulting rule
Questions about this topic
Which form should be used when the roots are known?+
Use the factorised form shown above, where the roots determine the factors. A further point determines the scale
Which form should be used when the vertex is known?+
Use the vertex form shown above, where the vertex determines its position. A further point determines the scale
Why is a final substitution useful?+
It checks that the expanded rule still gives a stated output. This can reveal a sign or expansion error
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs
In this chapter
Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model