What Does a Vertical Compression Look Like?
A vertical compression is a transformation that “squeezes” the graph of a function toward the x‑axis, making it appear flatter while preserving its horizontal position. But the x‑coordinates stay unchanged, so the shape of the graph is retained but its height is reduced. In practice, when you apply a vertical compression, every y‑coordinate of the original graph is multiplied by a constant factor c where 0 < c < 1. Below you will find a detailed description of what a vertical compression looks like, how to recognize it, and how to apply it to various functions.
Understanding Vertical Compression
In the language of function transformations, a vertical compression is expressed as
[ g(x) = c \cdot f(x) ]
where f(x) is the original function and c is the compression factor. If c = ½, the new graph is exactly half as tall as the original; if c = 0.2, the graph is compressed to 20 % of its original height. The key visual cue is that the graph moves closer to the x‑axis without shifting left or right.
Visual Characteristics
- Flattened Appearance: Peaks and valleys become less pronounced.
- Same X‑Intercepts: Points where the graph crosses the x‑axis remain unchanged because multiplying zero by any factor still yields zero.
- Preserved Shape: The overall pattern (e.g., the sinusoidal wave of a sine function) stays recognizable; only the amplitude shrinks.
- No Reflection: Unlike a vertical reflection (‑f(x)), a compression does not flip the graph over the x‑axis; it merely reduces magnitude.
How to Identify a Vertical Compression
If you're compare two graphs, look for these tell‑tale signs:
- Equal X‑Coordinates: Corresponding points on the two graphs share the same x‑value.
- Reduced Y‑Magnitude: For any given x, the y‑value of the transformed graph is smaller in absolute value than that of the original.
- Constant Ratio: The ratio y₂ / y₁ ( transformed y over original y ) is the same for all points where y₁ ≠ 0. This ratio equals the compression factor c.
- Unchanged X‑Intercepts and Asymptotes: Vertical asymptotes, holes, and x‑intercepts stay in place.
If you notice that the graph looks “squashed” toward the x‑axis while keeping its horizontal layout, you are observing a vertical compression.
Mathematical Representation
General Formula
[ g(x) = c \cdot f(x) \quad \text{with} \quad 0 < c < 1 ]
- c = 1 → No change (identity transformation).
- 0 < c < 1 → Vertical compression.
- c > 1 → Vertical stretch (the opposite effect).
- c < 0 → Combination of compression/stretch and reflection across the x‑axis.
Effect on Key Features
| Feature | Original f(x) | After g(x) = c·f(x) |
|---|---|---|
| Y‑intercept | (0, f(0)) | (0, c·f(0)) |
| X‑intercept(s) | (x₀, 0) | (x₀, 0) – unchanged |
| Maximum/Minimum | y = ±A | y = ±c·A |
| Period (for periodic functions) | T | T – unchanged |
| Asymptotes (vertical) | x = a | x = a – unchanged |
| Asymptotes (horizontal) | y = L | y = c·L – scaled |
Real talk — this step gets skipped all the time Easy to understand, harder to ignore..
Examples with Different Functions
1. Linear Function
Original: f(x) = 2x + 3
Compressed (c = 0.4·(2x + 3) = 0.And 4): g(x) = 0. 8x + 1.
- The slope drops from 2 to 0.8, making the line less steep.
- The y‑intercept moves from 3 to 1.2, but the x‑intercept shifts from –1.5 to –1.5 (still the same because solving 0.8x + 1.2 = 0 gives x = –1.5).
2. Quadratic Function
Original: f(x) = x² − 4
Compressed (c = 0.In real terms, 3): g(x) = 0. 3·(x² − 4) = 0.3x² − 1.
- The parabola widens vertically; its vertex moves from (0, −4) to (0, −1.2).
- The x‑intercepts at x = ±2 remain unchanged because 0.3·(±2)² − 1.2 = 0.
3. Trigonometric Function
Original: f(x) = sin(x)
Compressed (c = 0.5): g(x) = 0.5·sin(x)
- Amplitude reduces from 1 to 0.5.
- The period (2π) and the x‑intercepts (multiples of π) stay the same.
- The wave looks “flatter” but retains its sinusoidal shape.
4. Rational Function
Original: f(x) = 1/(x − 2)
Compressed (c = 0.2): g(x) = 0.2/(x − 2)
- The vertical asymptote at x = 2 is unchanged.
- The graph approaches the asymptote more slowly; values are closer to zero for the same x.
Step‑by‑Step Guide to Apply a Vertical Compression
- Identify the Original Function f(x).
- Choose a Compression Factor c such that 0 < c < 1.
- Multiply Every Output by c: compute g(x) = c·f(x).
- Adjust Key Points:
- Multiply each y‑coordinate of notable points (intercepts, maxima, minima) by c.
- Keep x‑coordinates unchanged.
- Redraw the Graph:
- Plot the transformed points.
- Sketch the curve preserving the original shape but with reduced height.
- Verify:
- Check that x‑intercepts and vertical asymptotes remain in place.
Common Pitfalls to Avoid
When performing vertical compressions, students often make several common errors. Being aware of these can help ensure accuracy:
- Confusing Vertical with Horizontal Transformations: A vertical compression affects the y-values (outputs), whereas a horizontal compression affects the x-values (inputs). To give you an idea, $g(x) = 0.5f(x)$ is a vertical compression, but $g(x) = f(0.5x)$ is actually a horizontal stretch.
- Applying the Factor to the $x$-coordinate: Always remember that $c$ is being multiplied by the entire function. This means you only scale the $y$-coordinate. If you multiply the $x$-coordinate by $c$, you are performing a horizontal transformation instead.
- Forgetting the Distributive Property: If the function is written in a form like $f(x) = x^2 + 4$, a vertical compression of $c = 0.5$ must be applied to the entire expression: $0.5(x^2 + 4) = 0.5x^2 + 2$. A common mistake is to only multiply the first term, resulting in $0.5x^2 + 4$, which is incorrect.
- Misinterpreting the $x$-intercepts: While the $y$-values change, the $x$-intercepts are the "anchors" of the graph. If your transformed graph has different $x$-intercepts than the original, an error has occurred.
Summary Table: Vertical Transformation Overview
To keep your studies organized, use this quick reference to distinguish between vertical stretches and compressions:
| Transformation Type | Mathematical Form | Condition | Visual Result |
|---|---|---|---|
| Vertical Stretch | $g(x) = c \cdot f(x)$ | $c > 1$ | Graph looks "taller" or steeper. |
| Vertical Compression | $g(x) = c \cdot f(x)$ | $0 < c < 1$ | Graph looks "shorter" or flatter. |
| Vertical Reflection | $g(x) = -f(x)$ | $c = -1$ | Graph flips over the x-axis. |
Conclusion
Understanding vertical compression is a fundamental skill in algebraic and trigonometric analysis. By multiplying the output of a function by a constant $c$ (where $0 < c < 1$), we effectively "squash" the graph toward the x-axis. This transformation alters the range and the specific y-values of the function—such as the amplitude of a sine wave or the vertex of a parabola—while leaving the domain and the x-intercepts untouched Most people skip this — try not to..
People argue about this. Here's where I land on it.
Mastering this concept allows you to predict how changes in coefficients will reshape a graph, providing a powerful tool for modeling real-world phenomena where intensity or magnitude is scaled down Most people skip this — try not to..
