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Worked solution

Combine a horizontal scale with a vertical translation

State a sequence of transformations which transforms the graph of y=f(x)y=f(x) to the graph of y=f(2x)+1y=f(2x)+1.

CJC · Transformations · Example 7

Method 1: scale, then translate

Answer

Scale parallel to the x-axis by factor 1/21/2, then translate 1 unit in the positive y-direction.

The Teacher emphasises the replacements in order: x by 2x2x, followed by y by y−1y-1. Each substitution applies to the current equation.

Halve horizontal distances

y=f(x)⟶y=f(2x)y=f(x)\quad\longrightarrow\quad y=f(2x)

The input multiplier is 2, so the horizontal factor is its reciprocal.

Translate upwards

y−1=f(2x)⟹y=1+f(2x)y-1=f(2x)\quad\Longrightarrow\quad y=1+f(2x)

Replace y by y−1y-1, then solve for y.

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Method 2: translate, then scale

Answer

Translate 1 unit in the positive y-direction, then scale parallel to the x-axis by factor 1/21/2.

The Teacher emphasises the reverse replacement order: y by y−1y-1, followed by x by 2x2x. These changes act on different coordinates, so their order can be exchanged without changing their sizes.

Translate upwards

y−1=f(x)⟹y=1+f(x)y-1=f(x)\quad\Longrightarrow\quad y=1+f(x)

The original output is increased by 1.

Halve horizontal distances

y=1+f(2x)y=1+f(2x)

Replace x throughout the current equation.

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