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Identify a plane transformation and its exact parameters

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When identifying a map from corresponding points, what should be compared first?

Year 11, VCE Units 1 & 2, Chapter 7, 4 difficulty levels, No calculatorBuild a worksheet

Identify a plane transformation

Compare corresponding coordinates across more than one pair before naming the transformation. Constant differences, sign or midpoint patterns, and positive ratios point to different transformation families

\[\mathbf{p}'=M\mathbf{p}+\mathbf{t}\]
  1. Compare coordinate differences For a translation, subtract each pre-image coordinate from the matching image coordinate and look for one repeated vector \(\displaystyle \mathbf{p}'-\mathbf{p}=\mathbf{t}\)
  2. Check signs and midpoints For reflection in a line parallel to a coordinate axis, the perpendicular coordinate changes, and the reflecting line passes through the midpoint of each point and its image \(\displaystyle (x,y)\mapsto(x,-y),\quad c=\frac{y+y'}{2}\)
  3. Compare positive ratios A dilation has a constant positive ratio on each affected coordinate; an unchanged coordinate has ratio 1 \(\displaystyle x'=k_xx,\quad y'=k_yy,\quad k_x,k_y>0\)
  4. Check a second pair and sequence Use another mapped pair to confirm the parameters, and keep the order of component maps in a combination \(\displaystyle T_2(T_1(\mathbf{p}))\)

Worked example

The points P, Q and R map as shown. Identify the transformation and state its translation vector

\[P=(-1,2)\mapsto P'=(3,2),\quad Q=(2,-3)\mapsto Q'=(6,-3),\quad R=(0,1)\mapsto R'=(4,1)\]
  1. \(\displaystyle t_x=3-(-1)=4,\quad t_y=2-2=0\)The first pair gives a horizontal change of 4 and no vertical change
  2. \(\displaystyle 6-2=4,\quad -3-(-3)=0\)The second pair has the same coordinate differences, so the map is consistent
  3. \(\displaystyle \mathbf{p}'=\mathbf{p}+\begin{pmatrix}4\\0\end{pmatrix}\)A constant image-minus-pre-image vector identifies a translation
  4. \(\displaystyle R=(0,1)\mapsto R'=(0+4,1+0)=(4,1)\)The third pair verifies the same translation vector on another point

Common mistakes

Reversing the coordinate difference
✕\(\displaystyle t_x=x-x'=(-2)-3=-5\)✓\(\displaystyle t_x=x'-x=3-(-2)=5\)
For a forward map, the translation is image minus pre-image in each coordinate
Calling a translation an origin-fixing matrix map
✕\(\displaystyle \mathbf{p}'=M\mathbf{p},\quad\mathbf{t}=\begin{pmatrix}0\\0\end{pmatrix}\)✓\(\displaystyle \mathbf{p}'=\mathbf{p}+\mathbf{t},\quad\mathbf{t}=\begin{pmatrix}5\\-3\end{pmatrix}\)
The same non-zero displacement is added to every point, so the origin is not fixed and the vector must be shown separately
Using one factor for a two-axis dilation
✕\(\displaystyle x'=2x,\quad y'=y\)✓\(\displaystyle x'=2x,\quad y'=3y\)
A two-axis dilation can have a separate positive factor on each coordinate, so both factors must be identified

Difficulty levels

introName a standard axis transformation from exact coordinate data
standardRecover a translation component or one-axis dilation factor from mapped points
challengeDistinguish an offset reflection from a two-axis dilation using exact data
examIdentify a simple combination and its order from several representations

Questions about this topic

How can a translation be recognised from mapped points?+

Subtract each pre-image point from its image point. If the same vector results for every pair, the map is a translation by that vector

How can a reflection in a line parallel to an axis be identified?+

The affected coordinate changes sign about a fixed midpoint. The midpoint of that coordinate and its image gives the coordinate of the reflecting line

Why should a second mapped pair be checked?+

One pair can fit several possible rules, while a second pair tests whether the proposed parameters work consistently

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 7: Matrices and plane transformations

In this chapter

Connect a transformed rule, graph and coordinate mapWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrixSolve a pair of simultaneous linear equationsMap or recover points and finite coordinate setsIdentify a plane transformation and its exact parametersState the order of a matrix and locate its elementsAdd and subtract matricesMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matricesIdentify and construct the identity matrixFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixCompose two plane transformations in orderEvaluate powers of a square matrixDetermine the image rule of a basic function or relation