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Horizontal Shrink — Definition, Formula & Examples

Horizontal Shrink
Horizontal Compression

A shrink in which a plane figure is distorted horizontally.

 

 

See also

Vertical shrink

Key Formula

g(x)=f(cx)where c>1g(x) = f(cx) \quad \text{where } c > 1
Where:
  • f(x)f(x) = The original function before the transformation
  • g(x)g(x) = The transformed function after the horizontal shrink
  • cc = The compression factor; because c > 1, every x-value is divided by c, shrinking the graph horizontally by a factor of 1/c

Worked Example

Problem: The function f(x) = x² is transformed into g(x) = (3x)². Describe the transformation and find g(x) at x = 1 and x = 2.
Step 1: Identify the form of the transformation. Here g(x) = f(3x), so c = 3.
g(x)=f(3x)=(3x)2g(x) = f(3x) = (3x)^2
Step 2: Determine the shrink factor. Because c = 3, the graph is compressed horizontally by a factor of 1/3. Every point on the original graph moves to one-third of its original horizontal distance from the y-axis.
Horizontal shrink factor=1c=13\text{Horizontal shrink factor} = \frac{1}{c} = \frac{1}{3}
Step 3: Evaluate g(x) at x = 1. On the original f, f(1) = 1. On g, the same output of 9 that originally occurred at x = 3 now occurs at x = 1.
g(1)=(3⋅1)2=9g(1) = (3 \cdot 1)^2 = 9
Step 4: Evaluate g(x) at x = 2. The output of 36 that originally occurred at x = 6 on f now occurs at x = 2.
g(2)=(3⋅2)2=36g(2) = (3 \cdot 2)^2 = 36
Answer: The graph of f(x) = x² is horizontally shrunk by a factor of 1/3 to produce g(x) = (3x)² = 9x². Points reach the same y-values at one-third the x-distance: g(1) = 9 and g(2) = 36.

Another Example

Problem: Given f(x) = sin(x), graph g(x) = sin(2x) and explain how it relates to f.
Step 1: Write g in transformation form: g(x) = f(2x), so c = 2.
g(x)=sin⁡(2x)g(x) = \sin(2x)
Step 2: The horizontal shrink factor is 1/2. Every x-coordinate on the sine curve is halved.
Shrink factor=12\text{Shrink factor} = \frac{1}{2}
Step 3: The original sine function has period 2π. After the shrink, the period becomes π.
New period=2π2=π\text{New period} = \frac{2\pi}{2} = \pi
Answer: g(x) = sin(2x) is a horizontal shrink of sin(x) by a factor of 1/2, cutting the period in half from 2π to π. The amplitude stays at 1.

Frequently Asked Questions

How do you tell if a transformation is a horizontal shrink or a horizontal stretch?
Look at the constant c multiplying x inside the function, written as f(cx). If c > 1, the graph shrinks horizontally (compresses toward the y-axis). If 0 < c < 1, the graph stretches horizontally (expands away from the y-axis). The actual scale factor applied to x-coordinates is 1/c.
Why does multiplying x by a number greater than 1 shrink the graph instead of stretching it?
It seems counterintuitive, but multiplying x by c > 1 inside f(cx) means the function reaches any given output value sooner — at a smaller x. For example, f(2x) at x = 3 gives the same result as f(x) at x = 6. Each point moves closer to the y-axis, which compresses the graph.

Horizontal Shrink vs. Vertical Stretch

A horizontal shrink (g(x) = f(cx), c > 1) compresses the graph toward the y-axis by reducing x-coordinates, making the shape narrower. A vertical stretch (g(x) = c · f(x), c > 1) pulls the graph away from the x-axis by multiplying y-coordinates, making the shape taller. Both can make a graph look 'steeper,' but they act on different axes. A horizontal shrink changes the input; a vertical stretch changes the output.

Why It Matters

Horizontal shrinks appear whenever you change the rate or frequency of a process. In trigonometry, multiplying the input of sin(x) by 2 doubles the frequency of a wave — essential in modeling sound, light, and alternating current. In data science and physics, rescaling the horizontal axis is a standard way to normalize measurements or compare phenomena that operate on different time scales.

Common Mistakes

Mistake: Thinking f(2x) stretches the graph horizontally by a factor of 2.
Correction: f(2x) actually shrinks the graph by a factor of 1/2. The multiplier c inside f(cx) is the reciprocal of the horizontal scale factor. Larger c means a narrower graph, not a wider one.
Mistake: Confusing a horizontal shrink with a vertical stretch because both can make a curve appear steeper.
Correction: Check where the constant appears. If it multiplies x inside the function, it is horizontal. If it multiplies the entire function output, it is vertical. For instance, (2x)² = 4x² looks like a vertical stretch by 4, but it originated as a horizontal shrink by 1/2 — they happen to produce the same result only for power functions.

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