Solved examples: describing a sequence of transformations geometry

📚 Quick Study Guide

• 🔍 Transformations: Transformations alter the position or size of a shape. Common transformations include translations, rotations, reflections, and dilations.
• ➡️ Translations: A translation 'slides' a shape. It's defined by a vector $(a, b)$, where $a$ is the horizontal shift and $b$ is the vertical shift.
• 🔄 Rotations: A rotation turns a shape around a fixed point (usually the origin). It's defined by an angle (e.g., 90°, 180°, 270°) and a direction (clockwise or counterclockwise).
• зеркало Reflections: A reflection 'flips' a shape over a line (e.g., the x-axis or y-axis). If reflecting over the x-axis, $(x, y) \rightarrow (x, -y)$. If reflecting over the y-axis, $(x, y) \rightarrow (-x, y)$.
• 📐 Dilations: A dilation changes the size of a shape. It's defined by a scale factor $k$. If $k > 1$, the shape gets bigger. If $0 < k < 1$, the shape gets smaller. The coordinates are multiplied by $k$: $(x, y) \rightarrow (kx, ky)$.
• 📝 Sequence of Transformations: Applying multiple transformations one after another. Order matters!

Practice Quiz

1. Question 1: A point (2, 3) is translated by the vector (-1, 2). What are the coordinates of the image?
   1. (3, 5)
   2. (1, 5)
   3. (1, 1)
   4. (3, 1)
2. Question 2: A triangle has vertices A(1, 1), B(2, 3), and C(4, 1). It is reflected over the x-axis. What are the coordinates of B'?
   1. (2, 3)
   2. (-2, 3)
   3. (2, -3)
   4. (-2, -3)
3. Question 3: A point (4, -2) is rotated 90° counterclockwise about the origin. What are the coordinates of the image?
   1. (-2, -4)
   2. (2, 4)
   3. (2, -4)
   4. (-2, 4)
4. Question 4: A square with vertices (0, 0), (0, 2), (2, 2), and (2, 0) is dilated by a scale factor of 0.5. What is the area of the new square?
   1. 4
   2. 1
   3. 2
   4. 0.5
5. Question 5: A point (1, 4) is first reflected over the y-axis and then translated by (2, -1). What are the final coordinates?
   1. (3, 3)
   2. (1, 3)
   3. (-1, 3)
   4. (-3, -5)
6. Question 6: Which transformation does NOT preserve the size and shape of the figure?
   1. Translation
   2. Rotation
   3. Reflection
   4. Dilation
7. Question 7: A triangle is rotated 180° about the origin, then reflected over the y-axis. If the original point A is (3, -2), what is the final image A''?
   1. (3, -2)
   2. (-3, 2)
   3. (-3, -2)
   4. (3, 2)