A vertical shrink by a factor of 1/2 transforms a graph so that every output value is multiplied by 1/2. The graph moves closer to the x-axis, becoming vertically compressed while its x-coordinates remain unchanged.
Introduction to Vertical Shrinks
A vertical shrink, also called a vertical compression, changes the distance between a graph and the x-axis. For a function written as
[ y=f(x), ]
a vertical shrink by a factor of 1/2 produces the function
[ g(x)=\frac{1}{2}f(x). ]
So in practice, each original output, or y-value, is divided by two. If a point on the original graph is ((x,y)), its corresponding point after the transformation is
[ \left(x,\frac{y}{2}\right). ]
The x-coordinate remains the same, but the y-coordinate becomes half as far from the x-axis Surprisingly effective..
Understanding the Factor 1/2
The factor (1/2) is a scale factor between 0 and 1. When a positive scale factor is less than 1, the graph is compressed vertically Small thing, real impact..
To give you an idea, suppose
[ f(3)=8. ]
After a vertical shrink by a factor of 1/2,
[ g(3)=\frac{1}{2}\cdot 8=4. ]
The input (x=3) has not changed. Only the output has changed from 8 to 4.
Consider the absolute value function (f(x)=|x|). Its vertex sits at the origin, and it rises with a slope of 1 on the right and (-1) on the left. Applying the shrink gives
[ g(x)=\frac{1}{2}|x|. ]
The V-shape remains, but each arm becomes half as steep. Where the original graph passed through ((2,2)), the transformed graph passes through ((2,1)). The vertex stays fixed at ((0,0)) because half of zero is still zero, so x-intercepts that lie on the x-axis are unchanged.
This behavior generalizes. Every y-coordinate is halved, which means the entire range of the function is scaled by (1/2). On the flip side, if the original range was ([-4,6]), the new range becomes ([-2,3]). If the original function had a maximum value of 10, the shrunken version peaks at 5. X-intercepts remain x-intercepts because (\frac{1}{2}\cdot 0 = 0), so points where the graph crosses the x-axis are fixed points of the transformation Simple, but easy to overlook. That's the whole idea..
It is worth distinguishing this from a horizontal shrink. A horizontal shrink by a factor of (1/2) would replace (x) with (
It is worth distinguishing this from a horizontal shrink. A horizontal shrink by a factor of ( \tfrac12 ) would replace (x) with (2x); that is,
[ g(x)=f(2x). ]
Now the input is “sped up.” For any point ((x,y)) on the original graph, the transformed point becomes
[ \left(\frac{x}{2},,y\right). ]
The y‑coordinates stay the same, while the x‑coordinates are halved, pulling the graph toward the y‑axis.
Comparing the Two Types of Shrink
| Transformation | Algebraic Form | Effect on Points ((x,y)) | Visual Effect |
|---|---|---|---|
| Vertical shrink (\displaystyle g(x)=\tfrac12 f(x)) | Multiply the output by (\tfrac12) | ((x,\tfrac{y}{2})) | Graph moves closer to the x‑axis; y‑values are reduced. |
| Horizontal shrink (\displaystyle g(x)=f(2x)) | Multiply the input by (2) | ((\tfrac{x}{2},y)) | Graph moves closer to the y‑axis; x‑values are reduced. |
Not the most exciting part, but easily the most useful.
Notice that the factor that appears in the algebraic expression is the reciprocal of the geometric scaling. A vertical shrink by (\tfrac12) uses a factor of (\tfrac12) directly, while a horizontal shrink by (\tfrac12) uses a factor of (2) inside the function argument.
Example: Absolute‑Value Function
Take the same absolute‑value function (f(x)=|x|).
Vertical shrink: (g(x)=\tfrac12|x|).
Points: ((2,2)\to(2,1)); the V‑shape is half as steep.
Horizontal shrink: (h(x)=|2x|).
Points: ((2,2)\to(1,2)); the V‑shape is mirrored but now reaches the same height at half the horizontal distance.
Both transformations preserve the x‑intercepts (the origin) because (|0|=0) and (\tfrac12\cdot0=0). On the flip side, they affect other features differently: the vertical shrink reduces the maximum and minimum values, whereas the horizontal shrink changes the width of the graph.
Combining Vertical and Horizontal Shrinks
When multiple shrinks are applied, the order matters only if the transformations are not commutative. To give you an idea, applying a vertical shrink by (\tfrac12) followed by a horizontal shrink by (\tfrac12) yields
[ k(x)=\tfrac12,f(2x). ]
If the order were reversed, the result would be the same because scalar multiplication of the output and scaling of the input are independent operations. In more complex scenarios involving shifts or stretches, careful sequencing becomes essential Not complicated — just consistent..
Why Understanding These Transformations Matters
Vertical and horizontal shrinks are fundamental tools for modeling real‑world phenomena where size changes are involved. In practice, in physics, they describe how waveforms compress under different media; in economics, they illustrate how quantities scale with price changes; in computer graphics, they enable the resizing of images without distortion. Mastering these transformations equips students with the ability to predict and manipulate the shape of graphs across disciplines.
Conclusion
A vertical shrink by a factor of (\tfrac12) compresses a graph toward the x‑axis by halving every y‑value while leaving x‑coordinates untouched. The algebraic signature is a simple multiplication of the original function by (\tfrac12), and the geometric effect is a reduction of the graph’s vertical extent. By contrasting this with a horizontal shrink—where the input variable is multiplied by the reciprocal factor—we gain a fuller picture of how functions can be scaled in different directions. These insights are not only central to algebraic graphing but also provide a foundation for applications ranging from scientific modeling to digital design.
