Quadratic transformations and features
Completed-square form shows how a parabola is changed from the base quadratic. The same form gives the turning point, axis of symmetry and concavity, while horizontal intercepts are found by setting the height to zero
\[y=a(x-h)^{2}+k\]
- Read completed-square form Use the completed-square form shown here to identify the coefficient and the two translations \(\displaystyle y=a(x-h)^{2}+k\)
- Interpret the coefficient A negative a reflects the graph in the x-axis, and a larger absolute value makes it narrower \(\displaystyle a<0\Rightarrow\text{reflection},\qquad |a|>1\Rightarrow\text{narrower}\)
- Read the vertex and axis The turning point is (h,k), and the vertical axis of symmetry is x = h \(\displaystyle y=2(x-3)^{2}+4\Rightarrow V=(3,4),\quad x=3\)
- Find horizontal intercepts Set y equal to zero and solve for the x-values where the graph meets the horizontal axis \(\displaystyle y=(x-2)(x-5)\Rightarrow x=2\text{ or }x=5\)
Worked example
For the quadratic below, describe the transformations from the base quadratic and state the turning point and axis of symmetry
\[g(x)=-2(x+1)^{2}+8\]
- \(\displaystyle a=-2,\qquad h=-1,\qquad k=8\)Read the three parameters from completed-square form
- \(\displaystyle a<0\Rightarrow\text{reflection in the x-axis},\qquad |a|=2\Rightarrow\text{vertical stretch by factor }2\)The sign gives the reflection and the absolute value gives the vertical dilation
- \(\displaystyle h=-1\Rightarrow\text{translate 1 unit left},\qquad k=8\Rightarrow\text{translate 8 units up}\)The bracket x + 1 is x - (-1), so the horizontal translation is left by 1
- \(\displaystyle V=(-1,8),\qquad x=-1\)The vertex is (h,k) and the axis of symmetry is the line x = h
Common mistakes
Reversing the horizontal sign
✕\(\displaystyle y=2(x-3)^{2}+4\Rightarrow\text{translate 3 units left}\)✓\(\displaystyle y=2(x-3)^{2}+4\Rightarrow\text{translate 3 units right}\)
The sign inside x - h is opposite to the direction of the horizontal translationReversing the width effect
✕\(\displaystyle y=3x^{2}\Rightarrow\text{the graph is wider than }y=x^{2}\)✓\(\displaystyle y=3x^{2}\Rightarrow\text{the graph is narrower than }y=x^{2}\)
Multiplying by a value with absolute value greater than 1 makes the parabola narrowerReading coefficients as intercepts
✕\(\displaystyle y=x^{2}-6x+8\Rightarrow x\text{-intercepts are }-6\text{ and }8\)✓\(\displaystyle y=x^{2}-6x+8=(x-2)(x-4)\Rightarrow x\text{-intercepts are }2\text{ and }4\)
The linear and constant coefficients are not the roots; set y to zero and solve the quadraticDifficulty levels
introDescribe one reflection, dilation or translation from completed-square form
standardDescribe combined transformations and find exact vertex and axis features
challengeConvert a general or factorised rule before finding its transformations and intercepts
examLink the input form, transformations, concavity, vertex, axis and intercept features
Questions about this topic
Why is the horizontal direction opposite to the bracket sign?+
The form uses x - h, so x + 3 means h = -3 and a translation 3 units left
What does the value of a change?+
The sign of a gives reflection in the x-axis, while the absolute value controls the vertical stretch or compression and therefore the width
How are horizontal intercepts found?+
Set y = 0 and solve for x; the resulting x-values are the points where the graph meets the horizontal axis
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