Porisma

Worked solution

Reverse a translation, reflection and vertical scale

A translates 2 units in the negative x-direction; B reflects in the x-axis; C scales parallel to y by 2 with the x-axis invariant. Applied in order A,B,C, the resulting equation is y=−2(x+3)/(x+2)y=-2(x+3)/(x+2). Find the equation before the transformations. marks

[2012/RI/Promo/2]

CJC · Review problems · Question 7

Answer

y=(x+1)/xy=(x+1)/x, x≠0.

Source method 1: reverse one step at a time

Before C,y=−(x+3)/(x+2)y=-(x+3)/(x+2). Before B,y=(x+3)/(x+2)y=(x+3)/(x+2). Before A, substitute x−2: y=[(x−2)+3]/[(x−2)+2]=(x+1)/xy=[(x-2)+3]/[(x-2)+2]=(x+1)/x.

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Source method 2: combine inverse substitutions

Undo the horizontal translation inside the final expression, divide by 2 to undo C and negate to undo B:

y=−(−2[(x−2)+3]2[(x−2)+2])=x+1x.\begin{aligned}&y\\ & =-\left(\frac{-2[(x-2)+3]}{2[(x-2)+2]}\right)\\ & =\frac{x+1}{x}.\end{aligned}

The original domain excludes x=0.

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Common mistakes

The original pole is x=0; the transformed pole x=−2 does not stay fixed under the inverse translation.

Solution notes

Both printed solution methods and every intermediate expression retained. No source figure.

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