Vertical Compression

Definition

Vertical compression is a transformation that scales the graph of a function vertically, either stretching or shrinking the graph along the y-axis. This transformation affects the amplitude or range of the function, altering the vertical scale without changing the horizontal scale.

5 Must Know Facts For Your Next Test

1. Vertical compression is a key concept in understanding and graphing logarithmic functions, as it affects the vertical scale and range of the function.
2. The vertical compression of a logarithmic function is controlled by the base of the logarithm, with a larger base resulting in a more compressed graph.
3. Vertical compression can be used to model real-world phenomena, such as the relationship between the magnitude of an earthquake and the energy released, or the decibel scale used to measure sound intensity.
4. Logarithmic functions with a base greater than 1 undergo vertical compression, while those with a base less than 1 undergo vertical expansion.
5. Vertical compression can be represented mathematically as $f(x) = a \cdot \log_b(x)$, where $a$ is the vertical scale factor and $b$ is the base of the logarithm.

Review Questions

• Explain how the base of a logarithmic function affects its vertical compression.
  • The base of a logarithmic function $f(x) = a \cdot \log_b(x)$ directly influences the degree of vertical compression. A larger base value $b$ results in a more compressed graph, as the logarithm function grows more slowly. Conversely, a smaller base value $b$ leads to a less compressed, or more expanded, graph. This is because the logarithm function with a smaller base grows more rapidly, increasing the vertical scale of the function.
• Describe the relationship between vertical compression and the amplitude of a logarithmic function.
  • Vertical compression of a logarithmic function $f(x) = a \cdot \log_b(x)$ is directly related to the amplitude of the function. The amplitude, which represents the vertical scale of the graph, is controlled by the scale factor $a$. A larger value of $a$ results in a greater amplitude and less vertical compression, while a smaller value of $a$ leads to a smaller amplitude and more vertical compression. The base $b$ of the logarithm also affects the vertical compression, with a larger base causing more compression and a smaller base causing less compression.
• Analyze how vertical compression can be used to model real-world phenomena involving logarithmic relationships.
  • Vertical compression of logarithmic functions is a crucial concept for modeling various real-world phenomena that exhibit logarithmic relationships. For example, the relationship between the magnitude of an earthquake and the energy released can be represented using a logarithmic function with vertical compression, where the base of the logarithm corresponds to the scale used to measure earthquake magnitude (e.g., the Richter scale). Similarly, the decibel scale used to measure sound intensity is a logarithmic scale with vertical compression, allowing for the representation of a wide range of sound levels. By understanding and applying the principles of vertical compression, researchers and analysts can effectively model and interpret these types of logarithmic relationships in the real world.

Related terms

Amplitude: The amplitude of a function is the distance between the midline and the maximum or minimum value of the function, representing the vertical scale of the graph.

Stretch: A vertical stretch is a transformation that increases the amplitude of a function, making the graph appear taller or wider along the y-axis.

Shrink: A vertical shrink is a transformation that decreases the amplitude of a function, making the graph appear shorter or narrower along the y-axis.