Apply a plane transformation
Use the coordinate rule on each point and keep the point labels in order. A translation uses a vector, while an origin-fixing transformation uses a matrix
\[\mathbf{p}'=\mathbf{p}+\mathbf{t}\quad\text{or}\quad\mathbf{p}'=M\mathbf{p}\]
- Write points as columns Place the x-coordinate above the y-coordinate and keep the point labels in their original order \(\displaystyle \mathbf{p}=\begin{pmatrix}x\\y\end{pmatrix}\)
- Add translations to every point A translation adds the same horizontal and vertical components to every point \(\displaystyle \mathbf{p}'=\mathbf{p}+\begin{pmatrix}a\\b\end{pmatrix}\Rightarrow(x',y')=(x+a,y+b)\)
- Reflect in the stated coordinate axis A reflection in an axis changes the coordinate perpendicular to that axis and leaves the parallel coordinate unchanged \(\displaystyle \text{in the x-axis: }(x,y)\mapsto(x,-y)\)
- Scale the affected coordinate A dilation from one axis multiplies the coordinate measured from that axis by a positive factor \(\displaystyle \text{from the y-axis: }(x,y)\mapsto(kx,y),\quad k>0\)
Worked example
Translate the given point by the given vector. Find the image point
\[\mathbf{p}=\begin{pmatrix}4\\-1\end{pmatrix},\quad\mathbf{t}=\begin{pmatrix}-2\\3\end{pmatrix}\]
- \(\displaystyle x'=4+(-2)=2\)Add the horizontal translation component to the x-coordinate
- \(\displaystyle y'=-1+3=2\)Add the vertical translation component to the y-coordinate
- \(\displaystyle P'=(2,2)\)The image point has the two calculated coordinates in the same order
Common mistakes
Reversing a translation
✕\(\displaystyle \mathbf{p}'=\mathbf{p}-\mathbf{t}=\begin{pmatrix}4\\-1\end{pmatrix}-\begin{pmatrix}-2\\3\end{pmatrix}=\begin{pmatrix}6\\-4\end{pmatrix}\)✓\(\displaystyle \mathbf{p}'=\mathbf{p}+\mathbf{t}=\begin{pmatrix}4\\-1\end{pmatrix}+\begin{pmatrix}-2\\3\end{pmatrix}=\begin{pmatrix}2\\2\end{pmatrix}\)
A forward translation adds the stated vector to the pre-image, so subtracting it reverses the directionReflecting the wrong coordinate
✕\(\displaystyle \text{Reflection in the x-axis: }(x,y)\mapsto(-x,y)\)✓\(\displaystyle \text{Reflection in the x-axis: }(x,y)\mapsto(x,-y)\)
Reflection in the x-axis changes the perpendicular y-coordinate and leaves the x-coordinate unchangedScaling only one coordinate in a uniform dilation
✕\(\displaystyle (x,y)\mapsto(2x,y)\)✓\(\displaystyle (x,y)\mapsto(2x,2y)\)
For the displayed uniform dilation, both coordinates must be multiplied by 2Difficulty levels
introApply one axis-aligned map to one integer point in the forward direction
standardApply one transformation to several points or recover a vector or factor
challengeRecover a pre-image or parameter for an offset reflection or rational dilation
examUse mixed coordinate data with a method-recognition or invariant-point part
Questions about this topic
Why must a translation be applied to every point in a coordinate set?+
A translation has one displacement vector, so every point receives the same horizontal and vertical changes while its label stays with the point
Which coordinate changes when reflecting in the x-axis?+
The y-coordinate changes sign because it is perpendicular to the x-axis; the x-coordinate is unchanged
How is a pre-image recovered from an image point?+
Undo the translation and then apply the exact inverse of the matrix transformation, or reverse the coordinate rule directly
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