Compression
Contraction
A transformation in which
a figure grows smaller.
Compressions may be with respect to a point (compression
of a geometric figure) or with respect to the axis of
a graph (compression
of a graph).
Note: Some high
school textbooks erroneously use the word dilation to
refer to all transformations in which the figure changes size,
whether the figure becomes larger or smaller. Compression (or contraction)
refers to transformations in which the figure becomes smaller.
Dilation properly refers only to transformations in which the figure
grows
larger.
Unfortunately the English language has no word that refers collectively
to both
stretching
and shrinking.
See
also
Dilation
of a geometric figure, dilation
of a graph
Worked Example
Problem: The parent function is f(x) = x². Write the equation of the graph after a vertical compression by a factor of 1/3, then sketch key points.
Step 1: Identify the type of compression and the scale factor. A vertical compression by a factor of 1/3 means you multiply the entire function by 1/3.
g(x)=31⋅f(x) Step 2: Substitute the parent function into the formula.
g(x)=31x2 Step 3: Compare key points. For the parent function, when x = 3, f(3) = 9. For the compressed function:
g(3)=31(9)=3 Step 4: Notice that every y-value is one-third of its original value. The point (3, 9) becomes (3, 3), and the point (6, 36) becomes (6, 12). The graph is "squished" toward the x-axis.
Original: (3,9)→Compressed: (3,3) Answer: The vertically compressed function is g(x) = (1/3)x². Every y-coordinate is multiplied by 1/3, pulling the parabola closer to the x-axis.
Another Example
This example shows horizontal compression (toward the y-axis), which is trickier because the multiplier goes inside the function argument and the scale factor is the reciprocal of what students might expect.
Problem: The parent function is f(x) = sin(x). Write the equation of the graph after a horizontal compression by a factor of 1/4.
Step 1: A horizontal compression by a factor of 1/4 means each x-value is multiplied by 1/4, which corresponds to replacing x with 4x in the function.
g(x)=f(4x) Step 2: Substitute the parent function.
g(x)=sin(4x) Step 3: Check the effect on the period. The parent function sin(x) has period 2π. For sin(4x), the period is compressed:
New period=42π=2π Step 4: Verify a key point. The parent function reaches its peak at x = π/2. The compressed function reaches its peak when 4x = π/2:
x=8π Answer: The horizontally compressed function is g(x) = sin(4x), with a period of π/2 instead of 2π. The wave completes its cycle four times faster.
Frequently Asked Questions
What is the difference between compression and dilation?
Compression makes a figure smaller (scale factor between 0 and 1 for vertical, greater than 1 inside the argument for horizontal), while dilation makes a figure larger. Many textbooks loosely use "dilation" for both, but strictly speaking, compression shrinks and dilation stretches. They are opposite transformations.
How do you tell if a graph is vertically or horizontally compressed?
A vertical compression has a multiplier between 0 and 1 outside the function: g(x) = a·f(x) where 0 < a < 1. A horizontal compression has a multiplier greater than 1 inside the function argument: g(x) = f(bx) where b > 1. Vertical compression pushes the graph toward the x-axis; horizontal compression pushes it toward the y-axis.
Does compression change the shape of a graph?
Compression changes how wide or tall a graph appears, but it preserves the basic shape. A parabola remains a parabola; a sine wave remains a sine wave. The proportions change, but features like symmetry, intercepts at the origin (if any), and general curve type stay the same.
Compression vs. Dilation (Stretch)
| Compression | Dilation (Stretch) |
|---|
| Definition | Transformation that makes a figure smaller | Transformation that makes a figure larger |
| Vertical formula | g(x) = a·f(x), where 0 < a < 1 | g(x) = a·f(x), where a > 1 |
| Horizontal formula | g(x) = f(bx), where b > 1 | g(x) = f(bx), where 0 < b < 1 |
| Effect on graph | Graph moves closer to the axis of compression | Graph moves farther from the axis of dilation |
| Scale factor | Between 0 and 1 (shrinks distances) | Greater than 1 (enlarges distances) |
Why It Matters
Compressions appear throughout algebra and precalculus whenever you transform parent functions—for instance, adjusting the amplitude of a sine wave or narrowing a parabola. In physics and engineering, compressions model real phenomena like damped oscillations where wave amplitudes shrink over time. Understanding compression is also essential for graphing transformations on standardized tests like the SAT and ACT.
Common Mistakes
Mistake: Confusing vertical and horizontal compression factors. Students often think g(x) = f(4x) stretches the graph horizontally because 4 is greater than 1.
Correction: A factor b > 1 inside the argument compresses the graph horizontally by a factor of 1/b. The x-coordinates get divided by b, making the graph narrower. Think of it as "the function reaches the same output values faster."
Mistake: Using a negative scale factor and calling it a compression.
Correction: Negative scale factors produce reflections, not compressions. A compression requires a positive factor. For example, g(x) = −(1/2)f(x) both compresses vertically and reflects across the x-axis—these are two separate transformations.
