Function Transformations • Topic 4 of 4

Compression

Multiplying a function by a constant between 0 and 1 compresses its graph vertically: a·f(x) with 0 < a < 1 makes every output a fraction of its former size, pushing the graph toward the x-axis so it looks shorter and wider. For example, y = (1/3)x² is a vertical compression of y = x² by a factor of 3 — each height is one-third as large. As with stretches, the x-intercepts stay fixed. A compression is the opposite of a stretch: a factor above 1 stretches, a factor between 0 and 1 compresses. The SAT tests recognising a vertical compression and its factor from an equation like (1/a)·f(x).

sat33t4 graphOVertical compressiony = (1/3)x²: compressed vertically (wider)y = x² (parent)transformed

✅ Solved examples

1. y = x² becomes y = (1/4)x². What transformation?
A vertical compression by a factor of 4.
2. y = |x| becomes y = (1/2)|x|. What transformation?
A vertical compression by a factor of 2.
3. Does multiplying a function by 1/5 stretch or compress it?
Compress (factor between 0 and 1).
4. A compression pushes the graph toward which axis?
The x-axis.

✏️ Practice — try these, take hints as needed

1. y = x² becomes y = (1/3)x². What transformation?
Factor between 0 and 1.
—
—
A vertical compression by a factor of 3.
2. y = |x| becomes y = (1/6)|x|. What transformation?
1/6 < 1.
—
—
A vertical compression by a factor of 6.
3. Multiplying a function by 1/4 produces a vertical:
< 1 → ?
—
—
Compression.
4. y = x³ becomes y = (1/2)x³. What transformation?
Factor 1/2.
—
—
A vertical compression by a factor of 2.
5. A vertical compression makes a graph look shorter and:
Pushed toward x-axis.
—
—
Wider.

📝 Topic test — 8 questions

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