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Apply and describe transformations of a function

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Transforming a function

A graph transformation can be tracked with one map for the input coordinate and one map for the output coordinate. Horizontal changes appear in the input of the rule in the reciprocal direction, while output changes stay outside the function

\[x'=s_xx+t_x,\quad y'=s_yy+t_y\]
  1. 1Map the coordinates separately Use the horizontal map for x and the vertical map for y. A negative vertical scale reflects the graph in the x-axis before any vertical translation \(\displaystyle x'=2x,\quad y'=-y\Rightarrow (x,y)\mapsto(2x,-y)\)
  2. 2Reverse a horizontal translation inside the rule A graph shift right by t uses x-t inside the function \(\displaystyle x'=x+4\Rightarrow g(x)=f\left(x-4\right)\)
  3. 3Use the reciprocal horizontal dilation factor A horizontal dilation by s maps x to sx, so the input of f is x divided by s. \(\displaystyle x'=3x\Rightarrow g(x)=f\left(\frac{x}{3}\right)\)
  4. 4Preserve the order of combined changes A dilation and a translation on the same axis can give different results when their order is changed \(\displaystyle 2\left(x+1\right)=2x+2\ne2x+1\)
Year 12, VCE Units 3 & 4, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Transforming a function

A graph transformation can be tracked with one map for the input coordinate and one map for the output coordinate. Horizontal changes appear in the input of the rule in the reciprocal direction, while output changes stay outside the function

\[x'=s_xx+t_x,\quad y'=s_yy+t_y\]
  1. Map the coordinates separately Use the horizontal map for x and the vertical map for y. A negative vertical scale reflects the graph in the x-axis before any vertical translation \(\displaystyle x'=2x,\quad y'=-y\Rightarrow (x,y)\mapsto(2x,-y)\)
  2. Reverse a horizontal translation inside the rule A graph shift right by t uses x-t inside the function \(\displaystyle x'=x+4\Rightarrow g(x)=f\left(x-4\right)\)
  3. Use the reciprocal horizontal dilation factor A horizontal dilation by s maps x to sx, so the input of f is x divided by s. \(\displaystyle x'=3x\Rightarrow g(x)=f\left(\frac{x}{3}\right)\)
  4. Preserve the order of combined changes A dilation and a translation on the same axis can give different results when their order is changed \(\displaystyle 2\left(x+1\right)=2x+2\ne2x+1\)

Worked example

The point P=(2,4) lies on a reference graph. It is transformed by a reflection in the y-axis, a translation 5 units to the right, a vertical dilation by factor 3 and a translation 2 units downward. Find the image point P'

\[x'=-x+5,\quad y'=3y-2\]
  1. \(\displaystyle x'=-x+5\)The reflection and horizontal translation determine the horizontal coordinate map
  2. \(\displaystyle x'=-2+5=3\)Substitute the original x-coordinate 2 into the x-map
  3. \(\displaystyle y'=3y-2\)The vertical dilation and downward translation determine the vertical coordinate map
  4. \(\displaystyle y'=3(4)-2=10\Rightarrow P'=(3,10)\)Substitute the original y-coordinate 4 and combine the two image coordinates

Common mistakes

Swapping the coordinate maps
✕\(\displaystyle x'=2x-3,\quad y'=-y+4\Rightarrow P=(2,-1)\mapsto\left(-(-1)+4,\;2(2)-3\right)=(5,1)\)✓\(\displaystyle x'=2x-3,\quad y'=-y+4\Rightarrow P=(2,-1)\mapsto\left(2(2)-3,\;-(-1)+4\right)=(1,5)\)
The horizontal map must use the x-coordinate and the vertical map must use the y-coordinate
The horizontal dilation factor is not inverted
✕\(\displaystyle x'=2x\Rightarrow g(x)=f(2x)\text{ for a horizontal stretch by factor }2\)✓\(\displaystyle x'=2x\Rightarrow g(x)=f\left(\frac{x}{2}\right)\)
A horizontal dilation by factor 2 maps x to 2x, so the input of f is x divided by 2.
A horizontal translation keeps its direction inside f
✕\(\displaystyle g(x)=f\left(x+4\right)\text{ shifts the graph right by 4}\)✓\(\displaystyle g(x)=f\left(x-4\right)\text{ shifts the graph right by 4}\)
The input change is reversed inside the function: subtracting 4 from the input moves the graph right by 4

Difficulty levels

introMap one exact point after one translation or reflection
standardMap key points after one dilation and one translation
challengeHandle mixed transformations, undoing a rule and a non-commuting order comparison
examSelect a transformed description or rule and find two transformed key points

Questions about this topic

Which coordinate changes under a horizontal transformation?+

A horizontal transformation changes the input coordinate x. It does not directly change the output coordinate y.

Why does a horizontal dilation use a reciprocal inside the function?+

If a horizontal dilation multiplies each input coordinate by the scale factor, divide the new input by that factor to recover the original input

Why can the order of transformations matter?+

A dilation changes the size of a translation on the same axis. For example, translating by 1 then dilating by 2 gives a different result from dilating by 2 then translating by 1

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