Which Graph Shows A Dilation

Which Graph Shows a Dilation? A Comprehensive Guide to Identifying Similarity Transformations

Have you ever zoomed in on a digital map or resized a photograph without stretching or distorting it? If so, you’ve intuitively experienced the mathematical concept of dilation. In the world of graphing and coordinate geometry, a dilation is a specific type of transformation that changes the size of a figure but preserves its shape. This means all angles remain congruent, and all side lengths are multiplied by the same constant, known as the scale factor. The resulting figure is similar to the original. When presented with multiple graphs, identifying which one represents a true dilation requires a careful, methodical approach. This article will equip you with the knowledge and step-by-step strategies to confidently distinguish a dilation from other transformations like translations, rotations, or reflections.

Detailed Explanation: The Core Principles of Dilation

At its heart, a dilation is defined by two non-negotiable components: a center of dilation (a fixed point in the plane) and a scale factor (a positive real number, k). The transformation rule is elegantly simple: every point P in the original figure (the pre-image) is mapped to a new point P' such that P' lies on the ray emanating from the center C through P. The distance from C to P' is exactly k times the distance from C to P. Mathematically, if the center of dilation is the origin (0,0), the transformation is straightforward: the coordinates of any point (x, y) become (kx, ky). If the center is at some point (h, k), the formula adjusts to account for this shift: (x', y') = (k(x - h) + h, k(y - k) + k).

This process creates two critical outcomes. First, the shape is preserved. A triangle remains a triangle with identical angle measures; a rectangle remains a rectangle with right angles. Second, the size changes uniformly. If the scale factor k is greater than 1, the image is an enlargement. If k is between 0 and 1, the image is a reduction. A scale factor of exactly 1 results in no change at all—the figure is congruent to the original. The center of dilation is the only point that remains invariant; it maps directly onto itself. Understanding these principles is the foundation for analyzing any graph to determine if a dilation has occurred.

Step-by-Step Breakdown: How to Identify a Dilation on a Graph

When faced with a set of graphs showing an original figure and several transformed versions, follow this logical sequence to isolate the true dilation.

Step 1: Visually Assess Shape and Orientation. Begin by looking at the overall figures. Does the candidate image have the exact same shape as the pre-image? Are all angles clearly congruent? A dilation does not rotate, reflect, or shear the figure. The orientation (the "facing" direction) should be identical. If the image appears flipped, rotated, or skewed, it is not a pure dilation. For example, a right triangle should still have one 90-degree angle in the same relative corner.

Step 2: Identify Corresponding Points and Measure Distances from a Suspected Center. Select at least two, but preferably three or more, clearly corresponding vertices from the pre-image and the candidate image (e.g., point A and point A'). You must hypothesize or calculate the center of dilation. Often, especially in textbook problems, the center is the origin (0,0). If not obvious, you can find it by determining the intersection point of lines drawn through pairs of corresponding points (A to A', B to B', etc.). Once a potential center C is identified, measure the distance from C to a pre-image point and from C to its corresponding image point. Use the coordinate plane's grid to calculate