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Exploring Horizontal Stretch and Shrink by a Factor of 1/k: Mathematical Interpretation

August 18, 2023 by JoyAnswer.org, Category : Mathematics

What does a horizontal stretch or shrink by a factor of 1/k mean? Dive into the concept of horizontal stretch or shrink by a factor of 1/k in mathematical transformations. Understand the implications of this transformation on functions and geometric shapes.


Exploring Horizontal Stretch and Shrink by a Factor of 1/k: Mathematical Interpretation

What does a horizontal stretch or shrink by a factor of 1/k mean?

Horizontal stretch and shrink by a factor of 1/k is a transformation that affects the shape of a function's graph along the x-axis. This transformation is applied to a function in the form of y = f(kx), where k is a positive constant.

Horizontal Stretch and Shrink by a Factor of 1/k:

  1. Stretch (Compression) by a Factor of k: If k > 1, the graph of y = f(kx) is horizontally compressed (or stretched) compared to the graph of y = f(x). The graph is narrower and closer to the y-axis. The stretching factor is k.

  2. Shrink by a Factor of 1/k: If 0 < k < 1, the graph of y = f(kx) is horizontally stretched compared to the graph of y = f(x). The graph is wider and farther from the y-axis. The shrinking factor is 1/k.

Mathematical Interpretation:

Let's consider a function y = f(x). When we apply a horizontal stretch by a factor of k to this function, we substitute x with kx. The new function becomes y = f(kx). This means that for any x-coordinate on the original graph, we are now evaluating the function at kx, effectively compressing or stretching the graph horizontally.

Similarly, when we apply a horizontal shrink by a factor of 1/k, we substitute x with (1/k)x. The new function becomes y = f((1/k)x), which means that for any x-coordinate on the original graph, we are now evaluating the function at (1/k)x, effectively shrinking the graph horizontally.

Effect on Graph:

  • Stretch (Compression) by a Factor of k: The graph becomes narrower, and the points on the graph are closer to the y-axis. The horizontal distances between points are reduced by a factor of k, leading to a compressed appearance.

  • Shrink by a Factor of 1/k: The graph becomes wider, and the points on the graph are farther from the y-axis. The horizontal distances between points are increased by a factor of k, leading to a stretched appearance.

Example:Consider the function y = x^2. If we apply a horizontal stretch by a factor of 2 (k = 2), the new function becomes y = (2x)^2 = 4x^2. The graph is narrower compared to y = x^2. If we apply a horizontal shrink by a factor of 1/2 (k = 1/2), the new function becomes y = (1/2x)^2 = 1/4x^2. The graph is wider compared to y = x^2.

Horizontal stretch and shrink by a factor of 1/k is a powerful concept in mathematics and has applications in various fields, including physics, engineering, and economics. It allows us to manipulate and analyze functions to better understand their behavior.

Tags Horizontal Stretch/Shrink , Mathematical Concepts , Geometric Transformations

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