Unveiling Unit 9 Transformations Homework 5 Dilations: What Really Happened - A Step-by-Step Guide

This guide will walk you through understanding and completing Unit 9 Transformations Homework 5, focusing specifically on dilations. We'll break down the concepts, provide clear steps, and offer troubleshooting tips to help you conquer this assignment.

Prerequisites:

Before diving into this homework, ensure you have a basic understanding of the following:

  • Coordinate Plane: Familiarity with the x and y axes, plotting points, and identifying coordinates.

  • Basic Geometric Shapes: Understanding of triangles, squares, and other common shapes.

  • Transformations (General): A general understanding of what geometric transformations are – how they change the position or size of a shape.

  • Scale Factor: Know what a scale factor is (the multiplier used in dilations). If the scale factor is greater than 1, the shape gets bigger; if it's between 0 and 1, the shape gets smaller.

    • Tools You'll Need:

  • Pencil and Eraser: For accurate drawings and easy corrections.

  • Ruler: Essential for drawing straight lines and measuring distances.

  • Graph Paper (Recommended): Helps keep your drawings neat and accurate. Alternatively, you can print a graph paper template online.

  • Calculator (Optional): For calculations involving scale factors, especially if they are decimals or fractions.

  • Your Unit 9 Homework 5 Assignment: The actual problems you need to solve!

    • Step-by-Step Guide to Dilations:

    Let's break down the process of performing and understanding dilations, focusing on the key concepts that likely appear in your homework.

    Step 1: Identify the Center of Dilation:

    The center of dilation is the fixed point around which the shape is enlarged or reduced. This is *crucial*. The location of the center drastically affects the resulting dilated image.

  • Locate the Center: Your homework problem will usually explicitly state the center of dilation. It might be given as a coordinate point (e.g., (0, 0), (2, -1)) or described as "the origin."

  • Mark the Center: On your graph paper, clearly mark the center of dilation with a distinctive point (e.g., a small circle or a cross).

    • Step 2: Identify the Original Shape and Its Coordinates:

    Your homework will provide the coordinates of the original shape (also called the pre-image).

  • List the Coordinates: Carefully list the coordinates of each vertex (corner point) of the original shape. For example, if you have a triangle with vertices A, B, and C, you might have A(1, 1), B(3, 1), and C(2, 3).

  • Plot the Points: Plot these points on your graph paper and connect them to form the original shape. Use a pencil and ruler to ensure accurate lines.

    • Step 3: Determine the Scale Factor:

    The scale factor determines how much the shape will be enlarged or reduced.

  • Identify the Scale Factor: The problem will explicitly state the scale factor (e.g., 2, 0.5, 1/3).

  • Understand the Effect:

    • * Scale Factor > 1: The shape will be enlarged (made bigger).
    * Scale Factor < 1 (but > 0): The shape will be reduced (made smaller).
    * Scale Factor = 1: The shape remains the same size (no dilation).

    Step 4: Calculate the Coordinates of the Dilated Shape (Image):

    This is the core of the dilation process. You'll multiply the coordinates of each point of the original shape by the scale factor.

  • Multiply the x-coordinate: For each vertex, multiply the x-coordinate of the original point by the scale factor.

  • Multiply the y-coordinate: For each vertex, multiply the y-coordinate of the original point by the scale factor.

  • Example: If point A is (1, 1) and the scale factor is 2, the new point A' (A prime) will be (1 * 2, 1 * 2) = (2, 2).

    • Step 5: Plot the New Points and Draw the Dilated Shape:

    Now, plot the new coordinates you calculated in Step 4. These are the vertices of the dilated shape (also called the image).

  • Plot the Points: Plot each new point (A', B', C', etc.) on your graph paper.

  • Connect the Points: Use a ruler to connect the points and create the dilated shape. This shape will be similar to the original shape (same shape, different size).

    • Step 6: Verify Your Results:

  • Visual Inspection: Does the dilated shape appear to be the correct size relative to the original shape and the scale factor? Does it appear to be correctly positioned relative to the center of dilation?

  • Distance Check: Choose a point on the original shape and a corresponding point on the dilated shape. Measure the distance from the center of dilation to each of these points. The ratio of these distances should equal the scale factor. For example, if the scale factor is 2, the distance from the center of dilation to the dilated point should be twice the distance from the center to the original point.

    • Troubleshooting Tips:

  • Incorrect Center of Dilation: Double-check that you've correctly identified and marked the center of dilation. This is a common error.

  • Calculation Errors: Carefully review your multiplication of the coordinates by the scale factor. Use a calculator if needed.

  • Scale Factor Confusion: Remember that a scale factor between 0 and 1 reduces the size of the shape. A scale factor greater than 1 enlarges it.

  • Misinterpretation of Coordinates: Ensure you're correctly identifying the x and y coordinates of each point.

  • Negative Scale Factors (Advanced): Some problems might involve negative scale factors. A negative scale factor results in a dilation *and* a reflection across the center of dilation. Be extra careful with the signs when multiplying coordinates.

Example Problem:

Dilate triangle ABC with vertices A(1, 1), B(3, 1), and C(2, 3) using a scale factor of 2 and a center of dilation at the origin (0, 0).

1. Center of Dilation: (0, 0)
2. Original Coordinates: A(1, 1), B(3, 1), C(2, 3)
3. Scale Factor: 2
4. Calculate New Coordinates:
* A' = (1 * 2, 1 * 2) = (2, 2)
* B' = (3 * 2, 1 * 2) = (6, 2)
* C' = (2 * 2, 3 * 2) = (4, 6)
5. Plot and Draw: Plot A'(2, 2), B'(6, 2), and C'(4, 6) and connect them to form the dilated triangle.
6. Verify: The new triangle is twice the size of the original, and each point is twice as far from the origin as its corresponding original point.

Summary:

Dilation is a transformation that changes the size of a shape while maintaining its shape. The key to understanding dilations is understanding the center of dilation and the scale factor. By carefully multiplying the coordinates of the original shape by the scale factor and plotting the new points, you can accurately perform and understand dilations. Remember to double-check your calculations and visualize the transformation to ensure your answer makes sense. Good luck with your homework!