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Worked solution

Apply successive transformations in their stated order

(a) y=x2−1y=x^2-1 is translated 3 units in the positive x-direction, stretched by factor 2 parallel to y, then translated 2 units in the negative y-direction. Find the new equation as y=f(x)y=f(x). marks

(b) Three successive transformations change y=h(x)y=h(x) to y=h(1−2x)y=h(1-2x). Describe them. marks

[2011/CJC/MYE/11(modified)]

CJC · Review problems · Question 4

Answer

(a) y=2(x−3)2−4y=2(x-3)^2-4. (b) Left 1, horizontal scale 1/2, then reflection in y-axis.

(a) Translate horizontally

y=x2−1⟶y=(x−3)2−1y=x^2-1\longrightarrow y=(x-3)^2-1.

How did this part go?

(a) Stretch vertically, then translate down

A scale 2 parallel to y gives y/2=(x−3)2−1y/2=(x-3)^2-1, equivalently y=2(x−3)2−2y=2(x-3)^2-2. Translating down 2 gives y+2=2(x−3)2−2y+2=2(x-3)^2-2, hence y=2(x−3)2−4y=2(x-3)^2-4.

How did this part go?

(b) Retain the source three-step sequence

First translate 1 unit left: y=h(x+1)y=h(x+1). Next scale parallel to x by 1/2: y=h(2x+1)y=h(2x+1). Finally reflect in the y-axis: y=h(2(−x)+1)=h(1−2x)y=h(2(-x)+1)=h(1-2x).

How did this part go?

Common mistakes

Horizontal translations and scalings generally do not commute.

Solution notes

All source intermediate equations and the stated order retained. No source figure.

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