horizontal dilation example

HaC 2025

2 min read 18-03-2025

Horizontal dilation, a transformation in geometry, stretches or compresses a graph horizontally. This article will explore horizontal dilation with clear examples and explanations. We'll cover how to identify and perform this transformation. Understanding horizontal dilation is crucial for anyone working with functions and their graphical representations.

What is Horizontal Dilation?

Horizontal dilation scales a function's graph along the x-axis. A dilation factor, often represented by 'k', determines the extent of the stretch or compression. If |k| > 1, the graph stretches horizontally. If 0 < |k| < 1, the graph compresses horizontally. A negative value of 'k' also reflects the graph across the y-axis.

The Transformation Formula

The general formula for horizontal dilation is: y = f(kx)

f(x): The original function.
k: The dilation factor.

Examples of Horizontal Dilation

Let's illustrate horizontal dilation with several examples, starting with simple functions and progressing to more complex ones.

Example 1: Linear Function

Consider the function f(x) = x. Let's apply a horizontal dilation with k = 2.

The transformed function becomes: y = f(2x) = 2x

This doubles the slope of the original line, effectively compressing it horizontally. The y-intercept remains unchanged. The graph now appears steeper, although the change is technically a horizontal compression.

Example 2: Quadratic Function

Let's take the quadratic function f(x) = x². We'll apply a horizontal dilation with k = 1/2.

The transformed function is: y = f(1/2x) = (1/2x)² = (1/4)x²

The graph widens horizontally. The parabola stretches out along the x-axis. The vertex remains at the origin (0,0).

Example 3: More Complex Functions

Consider a more complex function like f(x) = sin(x). If we apply a horizontal dilation with k = 3, the transformed function becomes: y = sin(3x). This compresses the sine wave horizontally, increasing its frequency. The amplitude remains unchanged.

Example 4: Negative Dilation Factor

Let's use f(x) = x³ and apply a dilation with k = -1.

The transformed function is y = f(-x) = (-x)³ = -x³

This reflects the graph across the y-axis. The shape remains the same, but it's mirrored.

How to Identify Horizontal Dilation in a Graph

Given a graph, you can determine if horizontal dilation has been applied by observing:

Changes in the x-coordinates: If the x-coordinates are multiplied by a constant factor, it indicates horizontal dilation.
Stretching or Compression: A wider or narrower graph compared to the original function suggests horizontal dilation.
Reflection: If the graph is mirrored about the y-axis, it indicates a negative dilation factor.

Horizontal Dilation vs. Vertical Dilation

It's essential to differentiate horizontal dilation from vertical dilation. Vertical dilation scales the graph along the y-axis, affecting the y-coordinates. The formula for vertical dilation is: y = kf(x).

Conclusion

Understanding horizontal dilation is fundamental to working with functions and their transformations. By mastering the concept and the associated formula, you can accurately predict and interpret changes in a function's graph due to horizontal scaling. Remember to consider both the magnitude and the sign of the dilation factor (k) to fully understand the transformation. This knowledge is invaluable in various mathematical and graphical applications.