Horizontal Compression

Definition

Horizontal compression is a transformation of a function that alters the width or period of the function, without changing its height or amplitude. This term is relevant in the context of transforming functions, graphing logarithmic functions, and graphing trigonometric functions.

5 Must Know Facts For Your Next Test

1. Horizontal compression of a function $f(x)$ is achieved by replacing $x$ with $\frac{x}{a}$, where $a > 1$.
2. Horizontal compression results in the graph of the function becoming narrower, with the period of the function decreasing.
3. In the context of logarithmic functions, horizontal compression can be used to adjust the rate of growth or decay of the function.
4. For trigonometric functions, horizontal compression can be used to change the frequency or number of cycles within a given interval.
5. Horizontal compression is often used in combination with other transformations, such as vertical shifts or reflections, to create more complex transformations of functions.

Review Questions

• Explain how horizontal compression affects the graph of a function.
  • Horizontal compression of a function $f(x)$ results in the graph becoming narrower, with the period of the function decreasing. This is achieved by replacing $x$ with $\frac{x}{a}$, where $a > 1$. As the value of $a$ increases, the graph becomes more compressed horizontally, and the function completes more cycles within the same interval. This transformation can be used to adjust the rate of growth or decay in logarithmic functions, as well as the frequency or number of cycles in trigonometric functions.
• Describe how horizontal compression can be used to transform the graph of a logarithmic function.
  • In the context of logarithmic functions, horizontal compression can be used to adjust the rate of growth or decay of the function. By replacing $x$ with $\frac{x}{a}$, where $a > 1$, the graph of the logarithmic function becomes more compressed horizontally. This has the effect of increasing the rate of growth or decay, as the function reaches its asymptotic values more quickly. Horizontal compression of logarithmic functions can be a useful tool for modeling real-world phenomena that exhibit exponential growth or decay patterns.
• Analyze the impact of horizontal compression on the graph of a trigonometric function, and explain how it can be used to manipulate the function's characteristics.
  • Applying horizontal compression to a trigonometric function $f(x)$ by replacing $x$ with $\frac{x}{a}$, where $a > 1$, has the effect of changing the frequency or number of cycles within a given interval. As the value of $a$ increases, the graph of the trigonometric function becomes more compressed horizontally, and the function completes more cycles within the same interval. This transformation can be used to manipulate the characteristics of trigonometric functions, such as the number of oscillations or the rate at which the function reaches its maximum and minimum values. Horizontal compression of trigonometric functions is a valuable technique for modeling and analyzing periodic phenomena in various fields, such as physics, engineering, and biology.

Related terms

Dilation: A transformation that changes the scale of a function, either expanding or compressing it.

Period:

The distance between two consecutive repeating points on the graph of a periodic function.

Amplitude:

The maximum distance between the midline and the highest or lowest point of a periodic function.