91Ó°ÊÓ

Understanding the scale factor is pivotal when it comes to graphing dilations in transformation geometry. A scale factor, denoted here as \( k \), is the multiplier that determines how much a shape will be enlarged or reduced during a dilation.

In dilation, if the scale factor \( k > 1 \), the figure becomes larger; it's an enlargement. Conversely, if \( 0 < k < 1 \), the figure is reduced in size; it's a contraction. For the polygon with vertices X(6,-1), Y(-2,-4), Z(1,2), and a scale factor of \( k = 3 \), we face an enlargement.

To find the new coordinates of the dilated image, we multiply each coordinate of the original figure by the scale factor \( k \). In our exercise, we calculated the new position of each point by this scaling, resulting in the points (18,-3), (-6,-12), and (3,6) for X, Y, and Z respectively. It's essential to note that this transformation is uniform, meaning all points are affected equally, which keeps the polygon’s shape intact but alters its size.

Transformation geometry deals with the movement of figures in the plane. This involves shifting, rotating, reflecting, or dilating (resizing) shapes without altering their fundamental properties. Dilation is a form of transformation where figures are uniformly enlarged or reduced based on a scale factor.

Types of Transformations

• Translations: Slide a figure in a straight line from one position to another.
• Rotations: Turn a figure around a fixed point.
• Reflections: Flip a figure over a line to produce a mirror image.
• Dilations: Resize a figure proportionally from a central point.

In our current exercise, we focus on dilation, which does not change the shape but only the size of the figure. The process is straightforward: we apply the scale factor to each point's coordinates, effectively transforming the original polygon into a new, scaled version on the coordinate plane.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). It’s used to plot points, lines, and curves to represent graphical data. Each point on the plane is determined by an ordered pair of numbers, (x, y), which represent its horizontal and vertical positions respectively.

When graphing dilations, such as in the exercise we’re discussing, we use the coordinate plane to visualize the transformation. After computing the new coordinates by applying the scale factor, we plot both the original polygon and its dilated image on this plane. Connecting the plotted points demonstrates how the polygon has been transformed. For example, our original points X(6,-1), Y(-2,-4), Z(1,2) have been dilated to (18,-3), (-6,-12), (3,6) when we apply a scale factor of 3. Graphing these points showcases how the size of the figure has changed while maintaining its proportions and shape.