What Does a Vertical Compression Look Like: A Visual Guide
A vertical compression changes the shape of a graph by squeezing it closer to the x-axis. This transformation reduces the y-values of a function without shifting it up, down, left, or right. If you’re wondering what a vertical compression looks like, picture a parabola that appears wider and flatter or a sine wave that doesn’t rise or fall as high. The graph keeps its general shape and x-intercepts, but every point moves closer to the x-axis by a consistent factor. This effect is common in algebra and precalculus when studying how functions change.
What a Vertical Compression Looks Like
A vertical compression makes a graph appear “flattened” along the vertical axis. The transformation does not alter the x-values, so the graph doesn’t shift or stretch horizontally. Instead, all y-values are scaled down by multiplying the function by a number between 0 and 1.
For example, compare the graphs of $ f(x) = x^2 $ and $ g(x) = \frac{1}{2}x^2 $. The original function $ f(x) $ passes through the point $ (2, 4) $. After applying the compression, $ g(x) $ passes through $ (2, 2) $, since $ \frac{1}{2} \cdot 4 = 2 $. The same happens at $ (-2, 4) $, which becomes $ (-2, 2) $. The vertex remains at the origin, and the x-intercepts stay unchanged, but the entire curve rises more slowly. The parabola opens wider, visually confirming the compression.
This effect is not limited to quadratics. It applies to any function. For instance, the sine function $ f(x) = \sin(x) $ oscillates between -1 and 1. When compressed vertically by a factor of $ \frac{1}{3} $, the new function $ g(x) = \frac{1}{3}\sin(x) $ oscillates between $ -\frac{1}{3} $ and $ \frac{1}{3} $. The peaks and valleys are smaller, but the wave crosses the x-axis at the same points and repeats at the same intervals. The period is unchanged; only the amplitude is reduced.
The table below shows how specific outputs change under this transformation:
| $ x $ | $ f(x) = \sin(x) $ | $ g(x) = \frac{1}{3}\sin(x) $ |
|---|---|---|
| $ 0 $ | $ 0 $ | $ 0 $ |
| $ \frac{\pi}{2} $ | $ 1 $ | $ \frac{1}{3} $ |
| $ \pi $ | $ 0 $ | $ 0 $ |
| $ \frac{3\pi}{2} $ | $ -1 $ | $ -\frac{1}{3} $ |
| $ 2\pi $ | $ 0 $ | $ 0 $ |
Each y-value is one-third of the original. The x-intercepts remain at multiples of $ \pi $, but the maximum and minimum values shrink. This pattern is consistent across functions: vertical compression scales outputs uniformly without affecting inputs.
Identifying a Vertical Compression
To spot a vertical compression, compare the transformed graph to the original. Look for a reduction in height—peaks are lower, valleys are shallower—but no change in width or position. The x-intercepts stay the same, and the graph does not shift horizontally.
The presence of a coefficient between 0 and 1 in front of the function is a clear indicator. For example, $ y = 0.4f(x) $ means the function is compressed vertically by a factor of 0.4. At every x-value, the output is only 40% of what it was in the original. This coefficient is the scale factor.
Be careful not to confuse vertical compression with other transformations. A horizontal stretch spreads the graph out along the x-axis, changing the spacing between peaks in periodic functions. Vertical compression does not affect the x-axis spacing. The period, intercepts, and general timing of cycles remain unchanged.
Also, watch for reflections. If the coefficient is negative, such as in $ y = -0.5f(x) $, the graph is both compressed and flipped over the x-axis. The compression still applies—each y-value is halved in magnitude—but the graph opens downward instead of upward. Always check the sign and size of the multiplier to interpret the full transformation.
Mathematical Form and Behavior
A vertical compression is represented algebraically by multiplying the entire function by a constant $ a $, where $ 0 < a < 1 $. The general form is:
$$ g(x) = a \cdot f(x), \quad 0 < a < 1 $$
This equation scales every output value uniformly. The domain of the function stays the same, but the range becomes smaller. For bounded functions like sine and cosine, this means a smaller amplitude. For unbounded functions like quadratics or square roots, it results in a wider, less steep curve.
Take $ f(x) = \sqrt{x} $ as another example. Apply a vertical compression with $ a = 0.5 $ to get $ g(x) = 0.5\sqrt{x} $. At $ x = 4 $, $ f(4) = 2 $, but $ g(4) = 1 $. At $ x = 9 $, $ f(9) = 3 $, and $ g(9) = 1.5 $. The graph of $ g(x) $ rises more slowly, staying halfway between the x-axis and the original curve. The shape is preserved, but the vertical scale is reduced.
This point-by-point scaling applies across the entire domain. Key features like intercepts, asymptotes, and symmetry are maintained. However, slopes and curvature appear less steep, which can affect how the function behaves visually, especially when comparing rates of change.
Understanding vertical compression helps in analyzing real-world models where quantities scale down proportionally. For instance, adjusting the volume of a sound wave reduces its amplitude, which is a vertical compression of its waveform. In data modeling, scaling down a function can normalize values or fit curves to observed data.
For further study, resources like Khan Academy and Paul’s Online Math Notes offer clear explanations and practice problems on function transformations.
Key Takeaways
- A vertical compression makes a graph appear flatter by bringing all points closer to the x-axis.
- It occurs when a function is multiplied by a constant between 0 and 1, such as $ g(x) = 0.5f(x) $.
- The x-intercepts remain unchanged, but maximum and minimum values decrease in magnitude.
- You can identify it by comparing the height of the transformed graph to the original, especially at peaks and troughs.
- Be aware of reflections or horizontal stretches that may occur alongside compression.
- For further study on function behavior and graphing, resources like Khan Academy and Paul’s Online Math Notes offer reliable, in-depth material.
