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Interpret and determine a transformed power function

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Transformed power functions

The form y=a(x-h)^n+k shows the centre, exponent and scale of a transformed power function. Read those features first, then set y to zero when finding horizontal-axis intercepts

\[y=a\left(x-h\right)^n+k\]
  1. 1Read the centre In y=a(x-h)^n+k, the centre has coordinates (h,k) \(\displaystyle y=2(x-3)^4+1\Rightarrow(h,k)=(3,1)\)
  2. 2Use exponent parity For a nonzero scale a, an even integer exponent n ≥ 2 gives line symmetry and a turning point, while an odd integer exponent n ≥ 3 gives point symmetry and a stationary point of inflection \(\displaystyle n=4\Rightarrow\text{line symmetry about }x=h,\qquad n=5\Rightarrow\text{point symmetry about }(h,k)\)
  3. 3Use the scale sign The sign of a determines the orientation after the exponent parity is considered \(\displaystyle y=-3(x-1)^4+2\Rightarrow\text{both ends fall}\)
  4. 4Solve the intercept equation Set y equal to zero, isolate the power and use all real branches \(\displaystyle 0=2(x-1)^4-32\Rightarrow(x-1)^4=16\Rightarrow x=-1,\ 3\)
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Transformed power functions

The form y=a(x-h)^n+k shows the centre, exponent and scale of a transformed power function. Read those features first, then set y to zero when finding horizontal-axis intercepts

\[y=a\left(x-h\right)^n+k\]
  1. Read the centre In y=a(x-h)^n+k, the centre has coordinates (h,k) \(\displaystyle y=2(x-3)^4+1\Rightarrow(h,k)=(3,1)\)
  2. Use exponent parity For a nonzero scale a, an even integer exponent n ≥ 2 gives line symmetry and a turning point, while an odd integer exponent n ≥ 3 gives point symmetry and a stationary point of inflection \(\displaystyle n=4\Rightarrow\text{line symmetry about }x=h,\qquad n=5\Rightarrow\text{point symmetry about }(h,k)\)
  3. Use the scale sign The sign of a determines the orientation after the exponent parity is considered \(\displaystyle y=-3(x-1)^4+2\Rightarrow\text{both ends fall}\)
  4. Solve the intercept equation Set y equal to zero, isolate the power and use all real branches \(\displaystyle 0=2(x-1)^4-32\Rightarrow(x-1)^4=16\Rightarrow x=-1,\ 3\)

Worked example

Determine the transformed power rule with exponent 5 and centre (2,3) that passes through (3,5)

\[n=5,\qquad(h,k)=(2,3),\qquad(3,5)\text{ lies on the graph}\]
  1. \(\displaystyle y=a(x-2)^5+3\)Use the transformed power form with the given exponent and centre
  2. \(\displaystyle 5=a(3-2)^5+3\)Substitute the given point into the rule
  3. \(\displaystyle 2=a(1)^5\Rightarrow a=2\)Subtract 3 and divide by the translated power to find the scale
  4. \(\displaystyle y=2(x-2)^5+3\)The recovered rule has odd-power point symmetry about (2,3)

Common mistakes

Reversing the horizontal shift
✕\(\displaystyle y=2(x-3)^4-32\Rightarrow h=-3\)✓\(\displaystyle y=2(x-3)^4-32\Rightarrow h=3\)
The bracket is x-h, so x-3 gives h=3.
Ignoring exponent parity
✕\(\displaystyle n=4\Rightarrow\text{point symmetry and a stationary point of inflection}\)✓\(\displaystyle n=4\Rightarrow\text{line symmetry and a turning point}\)
An even exponent produces line symmetry and a turning point at the centre
Reversing the scale orientation
✕\(\displaystyle y=-2(x-1)^4+2\Rightarrow\text{both ends rise}\)✓\(\displaystyle y=-2(x-1)^4+2\Rightarrow\text{both ends fall}\)
A negative scale reflects an even-power graph so both ends fall away from the centre

Difficulty levels

introIdentify parity, centre and orientation from a canonical transformed power rule
standardCalculate horizontal intercepts and choose a graph for a translated or dilated power
challengeRecover a transformed power rule from its centre and one exact point or compare family members
examUse an inner-scaled rule to complete feature, graph and exact verification steps

Questions about this topic

What are h and k in y=a(x-h)^n+k?+

They give the centre coordinates (h,k). The horizontal coordinate is read with the opposite sign from the bracket when the bracket is written as x plus a number

How does the exponent affect the graph?+

For a nonzero scale a, even integer powers n ≥ 2 give line symmetry about x=h and a turning point at the centre. Odd integer powers n ≥ 3 give point symmetry about (h,k) and a stationary point of inflection

How are horizontal-axis intercepts found?+

Set y=0, isolate the power, and solve for every real branch allowed by the exponent and the isolated value

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