Horizontal Compression

Definition

Horizontal compression is a transformation of a function where the input values are scaled or compressed along the x-axis, resulting in a narrower or more condensed graph of the function. This compression affects the domain and rate of change of the function, while the range and general shape of the function remain unchanged.

5 Must Know Facts For Your Next Test

1. Horizontal compression of a function $f(x)$ is represented by the equation $f(a \cdot x)$, where $a$ is a positive constant greater than 1.
2. As the value of $a$ increases, the graph of the function becomes more compressed horizontally, with the domain becoming narrower.
3. Horizontal compression affects the rate of change of the function, as the function will have a steeper slope compared to the original function.
4. Horizontal compression can be used to model various real-world phenomena, such as the compression of sound waves in the Doppler effect.
5. The effects of horizontal compression are opposite to those of horizontal dilation, where the graph of the function becomes wider and the rate of change becomes less steep.

Review Questions

• Explain how horizontal compression affects the domain and rate of change of a function.
  • Horizontal compression of a function $f(x)$ is represented by the equation $f(a \cdot x)$, where $a$ is a positive constant greater than 1. As the value of $a$ increases, the graph of the function becomes more compressed horizontally, with the domain becoming narrower. This compression affects the rate of change of the function, as the function will have a steeper slope compared to the original function. The range and general shape of the function, however, remain unchanged.
• Describe the relationship between horizontal compression and horizontal dilation, and how they affect the properties of a function.
  • Horizontal compression and horizontal dilation are inverse transformations of a function. Horizontal compression, where the input values are scaled or compressed along the x-axis, results in a narrower or more condensed graph of the function. Horizontal dilation, on the other hand, is the opposite, where the input values are scaled or expanded along the x-axis, resulting in a wider graph of the function. The effects of these transformations are also opposite in terms of the rate of change, as horizontal compression leads to a steeper slope, while horizontal dilation results in a less steep slope.
• Analyze how horizontal compression can be used to model real-world phenomena, and explain the significance of this transformation in the context of 1.2 Basic Classes of Functions.
  • Horizontal compression can be used to model various real-world phenomena, such as the compression of sound waves in the Doppler effect. In the context of 1.2 Basic Classes of Functions, understanding horizontal compression is crucial as it represents a fundamental transformation that can be applied to different function classes, such as linear, quadratic, and exponential functions. Analyzing the effects of horizontal compression on the properties of these functions, including their domain, range, and rate of change, allows for a deeper understanding of how functions can be manipulated and applied to model and describe real-world situations.

Related terms

Dilation: The opposite of compression, where the input values are scaled or expanded along the x-axis, resulting in a wider graph of the function.

Vertical Compression: A transformation where the output values are scaled or compressed along the y-axis, resulting in a flatter graph of the function.

Transformation of Functions: The process of applying various operations, such as translation, reflection, dilation, and compression, to a function to obtain a new function with different properties.