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Determine the image rule of a basic function or relation

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To find an image rule after a coordinate transformation, what should be done first?

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

Find an image rule

An image rule is found from the coordinate change, not from a sampled graph point. Solve for the pre-image coordinates, substitute into the reference rule and then simplify

\[g(x)=s_yf\left(\frac{x-h_x}{s_x}\right)+h_y\]
  1. Write the coordinate equations Describe how each original coordinate becomes an image coordinate before changing the rule \(\displaystyle x'=s_xx+h_x,\quad y'=s_yy+h_y\)
  2. Solve for the pre-image coordinates Rearrange the coordinate equations so the original inputs are written in terms of the image coordinates \(\displaystyle x=\frac{x'-h_x}{s_x},\quad y=\frac{y'-h_y}{s_y}\)
  3. Substitute before simplifying Use the recovered original input in the reference rule, then apply the vertical scale and shift \(\displaystyle g(x')=s_yf\left(\frac{x'-h_x}{s_x}\right)+h_y\)
  4. Verify a mapped point Evaluate the reference rule at a known input and check that its image coordinates satisfy the new rule \(\displaystyle f(x)=x^{2}+3,\quad x'=2x+1,\quad y'=y-3,\quad(1,4)\mapsto(3,1),\quad g(3)=1\)

Worked example

Use the reference rule and coordinate change shown below to find the image rule

\[f(x)=2x+1,\quad x'=x+3,\quad y'=2y\]
  1. \(\displaystyle x'=x+3,\quad y'=2y\)These equations state the horizontal translation and vertical scale applied to each point
  2. \(\displaystyle x=x'-3,\quad y=\frac{y'}{2}\)Rearrange both equations to express the pre-image coordinates in terms of the image coordinates
  3. \(\displaystyle y'=2f(x'-3)=2\left(2(x'-3)+1\right)=4x'-10\)Substitute the recovered original input into f and apply the vertical scale
  4. \(\displaystyle g(x)=4x-10\)Rename the image input x after simplifying the image rule

Common mistakes

Using forward substitution
✕\(\displaystyle f(x)=x^{2},\quad x'=2x,\quad y'=y\Rightarrow g(x)=f(2x)=4x^{2}\)✓\(\displaystyle f(x)=x^{2},\quad x'=2x,\quad y'=y\Rightarrow g(x)=f\left(\frac{x}{2}\right)=\frac{x^{2}}{4}\)
The image coordinate is an output, so the original input must be recovered before it is substituted into the reference rule
Forgetting the vertical scale
✕\(\displaystyle f(x)=2x+1,\quad x'=x+3,\quad y'=2y\Rightarrow g(x)=f(x-3)=2x-5\)✓\(\displaystyle f(x)=2x+1,\quad x'=x+3,\quad y'=2y\Rightarrow g(x)=2f(x-3)=4x-10\)
The stated vertical change doubles every reference output after the horizontal input has been recovered
Reversing a horizontal translation
✕\(\displaystyle f(x)=2x+1,\quad x'=x+3,\quad y'=2y\Rightarrow g(x)=2f(x+3)=4x+14\)✓\(\displaystyle f(x)=2x+1,\quad x'=x+3,\quad y'=2y\Rightarrow g(x)=2f(x-3)=4x-10\)
Since the image input is the original input plus 3, the original input is the image input minus 3.

Difficulty levels

introFind a vertical translation or reflection of a monomial rule
standardFind the image rule after a horizontal or vertical change of a linear or quadratic rule
challengeFind the image rule after a two-way affine change or for a relation
examChoose the canonical image rule, fill exact coefficients and verify a mapped point

Questions about this topic

Why is a horizontal scale divided in the image rule?+

For a nonzero horizontal scale factor, divide the image input by the scale factor to recover the original input, then substitute that input into the reference rule

\[\begin{aligned}x'&=s_x x,\quad s_x\ne0\\x&=\frac{x'}{s_x}\end{aligned}\]
What does a vertical translation do to a function rule?+

With no other vertical change, add the translation amount to every reference output to obtain the image output

\[\begin{aligned}y'&=y+k\\g(x)&=f(x)+k\end{aligned}\]
How can an image rule be checked?+

Map a known point using the coordinate equations, then substitute its image x-coordinate into the image rule and compare the y-coordinate

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