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Connect a transformed rule, graph and coordinate map

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When connecting a transformed graph with its rule, what is a useful first check?

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

Interpret a transformed graph

Use exact coordinate changes to map a stable graph feature and then check another point. The graph, rule and coordinate map should all describe the same image

\[x'=s_xx+h_x,\quad y'=s_yy+h_y\]
  1. Map the horizontal coordinate Use the horizontal part of the coordinate map on the selected point or graph feature \(\displaystyle x'=s_xx+h_x\)
  2. Map the vertical coordinate Use the vertical part of the coordinate map separately from the horizontal part \(\displaystyle y'=s_yy+h_y\)
  3. Use a stable graph feature For a quadratic, the vertex gives a central point that can be mapped exactly \(\displaystyle y=a(x-h)^{2}+k\Rightarrow V=(h,k)\mapsto(s_xh+h_x,s_yk+h_y)\)
  4. Check a second point A second mapped point tests that the proposed transformation agrees with both the graph and the rule \(\displaystyle P=(x,y)\mapsto P'=(x',y'),\quad g(x')=y'\)

Worked example

Find the image vertex and image rule under the stated map

\[f(x)=-(x-2)^{2}+4,\quad x'=x-1,\quad y'=-y\]
  1. \(\displaystyle V=(2,4),\quad V'=(2-1,-4)=(1,-4)\)Map the vertex with the horizontal equation and then the vertical equation
  2. \(\displaystyle P=(3,3),\quad P'=(3-1,-3)=(2,-3)\)A second point maps consistently and provides a check on the graph transformation
  3. \(\displaystyle x'=x-1\Rightarrow x=x'+1,\quad y'=-y\Rightarrow y=-y'\)Solve the coordinate equations for the reference coordinates before forming the image rule
  4. \(\displaystyle g(x)=-f(x+1)=-\left(-(x-1)^{2}+4\right)=(x-1)^{2}-4\)Substitute the recovered input into the reference rule and apply the vertical reflection

Common mistakes

Assuming the origin is always invariant
✕\(\displaystyle x'=x-1,\quad y'=-y:\quad(0,0)\mapsto(0,0)\)✓\(\displaystyle x'=x-1,\quad y'=-y:\quad(0,0)\mapsto(-1,0)\)
A translation changes the origin unless both translation components are zero, so the origin must be mapped using the stated equations
Swapping the axis roles of a dilation
✕\(\displaystyle \text{from the y-axis by factor 2: }x'=x,\quad y'=2y\)✓\(\displaystyle \text{from the y-axis by factor 2: }x'=2x,\quad y'=y\)
A dilation from the y-axis changes the perpendicular x-coordinate and leaves the coordinate parallel to the axis unchanged
Using the forward input in the image rule
✕\(\displaystyle f(x)=x^{2},\quad x'=2x\Rightarrow g(x)=f(2x)=4x^{2}\)✓\(\displaystyle f(x)=x^{2},\quad x'=2x\Rightarrow g(x)=f\left(\frac{x}{2}\right)=\frac{x^{2}}{4}\)
The image input is twice the original input, so the original input is x divided by 2

Difficulty levels

introMatch a translated or reflected graph with its transformation description
standardRecover key image points and coefficients from the graph and reference rule
challengeIdentify a dilation or combination by comparing two transformed graphs
examReconcile the image rule, feature table and graph, including a counterexample distractor

Questions about this topic

Why is a vertex useful when interpreting a transformed quadratic graph?+

The vertex is a stable central feature with exact coordinates, so mapping it gives a direct comparison between the reference and image graphs

What is an invariant point?+

It is a point whose image is the same point after the transformation. The origin is invariant only when the stated map leaves both coordinates unchanged

Why should a second point be checked?+

A second point tests whether the coordinate map and proposed image rule agree beyond one selected feature

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