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Vertical Shrink — Definition, Formula & Examples

Vertical Shrink
Vertical Compression

A shrink in which a plane figure is distorted vertically.

 

 

See also

Horizontal shrink

Key Formula

g(x)=a⋅f(x),0<a<1g(x) = a \cdot f(x), \quad 0 < a < 1
Where:
  • f(x)f(x) = The original function
  • g(x)g(x) = The transformed (vertically shrunk) function
  • aa = The compression factor, a constant strictly between 0 and 1

Worked Example

Problem: The function f(x) = x² is vertically shrunk by a factor of 1/2. Write the new function g(x) and compare the y-values at x = 0, 2, and 4.
Step 1: Apply the vertical shrink by multiplying f(x) by 1/2.
g(x)=12⋅f(x)=12x2g(x) = \tfrac{1}{2} \cdot f(x) = \tfrac{1}{2}x^2
Step 2: Evaluate both functions at x = 0.
f(0)=0,g(0)=12(0)=0f(0) = 0, \quad g(0) = \tfrac{1}{2}(0) = 0
Step 3: Evaluate both functions at x = 2.
f(2)=4,g(2)=12(4)=2f(2) = 4, \quad g(2) = \tfrac{1}{2}(4) = 2
Step 4: Evaluate both functions at x = 4.
f(4)=16,g(4)=12(16)=8f(4) = 16, \quad g(4) = \tfrac{1}{2}(16) = 8
Step 5: Notice that every y-value of g is exactly half the corresponding y-value of f. The graph of g looks like a wider, flatter version of the parabola, pressed down toward the x-axis.
Answer: The vertically shrunk function is g(x) = (1/2)x². At x = 0, 2, and 4, the new y-values are 0, 2, and 8 — each exactly half of the original y-values 0, 4, and 16.

Frequently Asked Questions

How do you tell the difference between a vertical shrink and a vertical stretch?
Both transformations have the form g(x) = a · f(x). If 0 < a < 1, the graph is compressed toward the x-axis (vertical shrink). If a > 1, the graph is pulled away from the x-axis (vertical stretch). The value of the multiplier a is what determines which one you have.
Does a vertical shrink change the x-intercepts of a graph?
No. At any x-intercept, the y-value is 0, and multiplying 0 by any constant still gives 0. So all x-intercepts remain exactly where they were. However, every other point on the graph moves closer to the x-axis.

Vertical Shrink vs. Horizontal Shrink

A vertical shrink multiplies the output (y-values) by a factor between 0 and 1, compressing the graph toward the x-axis. A horizontal shrink multiplies the input (x-values) by a factor greater than 1 inside the function, compressing the graph toward the y-axis. For example, g(x) = (1/2)f(x) is a vertical shrink, while g(x) = f(2x) is a horizontal shrink. They affect different directions and follow different rules for the constant involved.

Why It Matters

Vertical shrinks appear whenever a real-world quantity is scaled down proportionally. For instance, if a sound wave's amplitude is reduced to 30% of its original level, the waveform undergoes a vertical shrink by a factor of 0.3. Understanding this transformation helps you quickly sketch modified graphs and interpret how changing a coefficient in a formula affects the shape of a curve.

Common Mistakes

Mistake: Confusing a vertical shrink with a horizontal stretch because both make the graph look 'wider' or 'flatter.'
Correction: Check where the constant appears. If it multiplies the entire function (outside), it is vertical. If it multiplies x inside the function argument, it is horizontal. The two transformations move points in different directions.
Mistake: Using a value of a greater than 1 and calling it a vertical shrink.
Correction: A vertical shrink requires 0 < a < 1. When a > 1, the transformation is a vertical stretch, which pulls the graph away from the x-axis rather than compressing it.

Related Terms