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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 11, VCE Units 1 & 2, Chapter 6, 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 shown 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 its image point

\[P=(2,4),\quad x'=-x+5,\quad y'=3y-2\]
  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
✕\(\displaystyle P=(2,-1)\mapsto\left(-(-1)+4,\;2(2)-3\right)=(5,1)\)✓\(\displaystyle P=(2,-1)\mapsto\left(2(2)-3,\;-(-1)+4\right)=(1,5)\)
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: x-4 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 the graph maps x to sx, the original input is x divided by s. Therefore the rule uses f(x/s)

Why can the order of transformations matter?+

A dilation changes the size of a translation on the same axis. For example, dilating after a translation gives 2(x+1), whereas translating after a dilation gives 2x+1.

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 6: Functions and relations

In this chapter

Classify a relation and state its domain and rangeEvaluate and determine function rulesFind 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 pointsDetermine a circle's equation and featuresAnalyse semicircles and circular regionsApply and describe transformations of a functionIntersections and tangents of a line and a circleEvaluate and interpret a restricted rational modelForm and analyse an inverse relation or function