Dilations: graph the image

Learning Objectives Define dilation, pre-image, image, center of dilation, and scale factor. Identify the center of dilation and the scale factor from a given problem or graph. Calculate the coordinates of a dilated image when the center of dilation is the origin. Graph the image of a polygon after a dilation on the coordinate plane. Distinguish between an enlargement and a reduction based on the scale factor. Calculate the coordinates of a dilated image when the center of dilation is not the origin. Apply dilations to solve simple geometric problems. Ever wondered how maps shrink entire cities onto a piece of paper, or how a projector makes a tiny image huge? 🗺️ These are all examples of transformations called dilations! In this lesson, you'll learn how to resize fi...

TermDefinitionExample DilationA transformation that changes the size of a figure, but not its shape. It either enlarges or reduces the figure.A small triangle becoming a larger triangle, but both having the same angles. Pre-imageThe original figure before a transformation is applied.If you start with triangle ABC, it is the pre-image. ImageThe new figure created after a transformation is applied. It is often denoted with prime notation (e.g., A'B'C').After dilating triangle ABC, the new triangle A'B'C' is the image. Center of DilationThe fixed point from which all points on the figure are stretched or shrunk. It's the point that doesn't move during the dilation.Often the origin (0,0) on a coordinate plane, but it can be any point. Scale Factor (k)Th...

Dilation Centered at the Origin $$(x, y) \to (kx, ky)$$ To find the coordinates of an image after a dilation centered at the origin (0,0), multiply both the x-coordinate and the y-coordinate of each vertex of the pre-image by the scale factor (k). Scale Factor Calculation $$k = \frac{\text{Length of Image Segment}}{\text{Length of Pre-image Segment}}$$ The scale factor (k) is the ratio of any corresponding side length in the image to its corresponding side length in the pre-image. It can also be found by dividing the coordinate of an image point by its corresponding pre-image point (e.g., $k = x'/x$). Identifying Enlargement or Reduction If $k > 1$, it's an enlargement. If $0 < k < 1$, it's a reduction. The value of the scale factor tells you...