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Worked solution

Two scalings of the same straight line

Given f(x)=x+1f(x)=x+1, sketch on separate diagrams (a) y=2f(x)y=2f(x) and (b) y=f(2x)y=f(2x).

CJC · Transformations · Self-Practice 2

(a) Sketch 2f(x)

Answer

y=2x+2y=2x+2, intercepts (−1,0)(-1,0) and (0,2)(0,2).

Independently derived: 2f(x)=2(x+1)=2x+22f(x)=2(x+1)=2x+2. Double every output, including the y-intercept; the x-intercept stays −1. Neither copy prints a self-practice solution here.

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Separate independently derived line sketches:2x+2 through (−1,0),(0,2), and 2x+1 through (−1/2,0),(0,1). Both slopes 2, distinct intercepts.
Separate independently derived line sketches:2x+2 through (−1,0),(0,2), and 2x+1 through (−1/2,0),(0,1). Both slopes 2, distinct intercepts.
Separate independently derived line sketches:2x+2 through (−1,0),(0,2), and 2x+1 through (−1/2,0),(0,1). Both slopes 2, distinct intercepts.
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(b) Sketch f(2x)

Answer

y=2x+1y=2x+1, intercepts (−1/2,0)(-1/2,0) and (0,1)(0,1).

Independently derived: f(2x)=2x+1f(2x)=2x+1. Halve input distances from the y-axis; the x-intercept moves to −1/2 and the y-intercept stays 1. Both new lines have gradient 2 but distinct y-intercepts. Draw separate diagrams as requested.

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