Dilation

Definition

Dilation is a transformation that alters the size of a figure while maintaining its shape and proportionality. This process involves expanding or contracting a figure based on a scale factor, which determines how much larger or smaller the figure will become. Dilation is essential in understanding geometric relationships and transformations, allowing for the manipulation of figures such as parabolas and circles, as well as their representation in function notation and graphing techniques.

5 Must Know Facts For Your Next Test

1. In dilation, every point of the original figure moves along a straight line away from or toward the center of dilation by a distance proportional to its distance from that center.
2. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction of the figure.
3. Dilation preserves the shape of geometric figures, meaning that angles remain unchanged, and the ratios of corresponding sides are consistent.
4. When dealing with functions, dilation can affect the vertical or horizontal stretch of the graph based on whether the scale factor is applied to the y-values or x-values.
5. In the context of parabolas, dilations can change the width of the parabola without altering its vertex position.

Review Questions

• How does dilation affect the dimensions and shape of geometric figures like parabolas and circles?
  • Dilation alters the dimensions of geometric figures by resizing them based on a scale factor while preserving their overall shape. For example, when a parabola undergoes dilation with a scale factor greater than 1, it stretches out and becomes wider, while a scale factor less than 1 compresses it and makes it narrower. Despite these changes in size, the relationships between points and angles within the figure remain consistent, ensuring that its fundamental properties are maintained.
• Explain how the concept of dilation connects to function notation and how it influences graphing transformations.
  • Dilation relates to function notation by affecting how functions are represented on graphs. When a function is transformed through dilation, its output values (y-values) can be multiplied by a scale factor, leading to vertical stretching or compressing. Additionally, changing input values (x-values) by using a scale factor results in horizontal transformations. Understanding how to apply these dilations helps visualize how functions behave differently when scaled up or down.
• Evaluate the importance of identifying the center of dilation in transformations and its effect on geometric figures.
  • Identifying the center of dilation is crucial because it determines how each point of a geometric figure moves during the transformation. The position of this center influences whether points are expanded outward or contracted inward. If the center is located outside the figure, points will move away from it, creating an enlargement; if it is inside, points will move toward it, resulting in a reduction. This understanding is key to accurately performing dilations and predicting how figures will transform.

Related terms

Scale Factor: The ratio that describes how much a figure is enlarged or reduced during dilation.

Transformation: A process that changes the position, size, or shape of a figure on a coordinate plane.

Center of Dilation:

The fixed point in space about which a figure is enlarged or reduced during the dilation process.