What is Dilation of a Graph?
Dilation of a graph is a transformation that changes the size of a graph. It's like using a magnifying glass to make the graph appear bigger or smaller. The key aspect of dilation is that it doesn't change the shape of the graph; it only scales it up or down.
Understanding Dilation
Imagine you have a graph of a simple function, like a line. To dilate this graph, we apply a scale factor. This scale factor can be a number greater than 1 (to enlarge the graph) or a fraction between 0 and 1 (to shrink the graph).
Here's how it works:
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Vertical Dilation: If you multiply the y-coordinates of all points on the graph by a scale factor, you're performing a vertical dilation. If the scale factor is greater than 1, the graph will be stretched vertically. If it's less than 1, the graph will be compressed vertically.
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Horizontal Dilation: Similarly, multiplying the x-coordinates of all points by a scale factor performs a horizontal dilation. A scale factor greater than 1 compresses the graph horizontally, while a scale factor less than 1 stretches it horizontally.
Example: Dilating a Line
Let's say we have a line with the equation y = x. We want to dilate this line vertically by a factor of 2. This means we multiply the y-coordinate of each point on the line by 2.
Original points: (1,1), (2,2), (3,3)
Dilated points: (1,2), (2,4), (3,6)
Notice how the line has become steeper, but the shape remains the same.
How to Identify Dilation
When looking at a graph, you can tell if it's been dilated by observing the following:
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Vertical Dilation: The distance between points on the y-axis will change proportionally to the scale factor. If the scale factor is 2, the distance between points on the y-axis will be doubled.
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Horizontal Dilation: The distance between points on the x-axis will change proportionally to the scale factor. If the scale factor is 1/2, the distance between points on the x-axis will be halved.
Applications of Dilation
Dilation is a fundamental transformation in mathematics and has many applications, including:
- Geometry: Dilation is used to create similar figures, where shapes have the same angles but different sizes.
- Computer Graphics: Dilation is essential for resizing and scaling images in computer graphics.
- Engineering: Dilation is used in scaling models and designs for different sizes.
- Physics: Dilation concepts are used in understanding the behavior of waves and other physical phenomena.
Conclusion
Dilation of a graph is a crucial transformation that allows us to change the size of a graph without altering its shape. By understanding the concept of scale factors and how they apply to vertical and horizontal dilation, we can perform these transformations accurately and confidently. Dilation finds applications in numerous fields, demonstrating its importance in various disciplines.