To understand the transformation from letter [tex]\( H ^{\prime \prime} \)[/tex] to letter [tex]\( H \)[/tex], follow these steps, keeping in mind the transformations provided:
### Step-by-Step Solution:
1. Translation:
- The letter [tex]\( H ^{\prime \prime} \)[/tex] is translated 2 units down. This means that every point of [tex]\( H ^{\prime \prime} \)[/tex] moves vertically downward by 2 units. Imagine placing the letter [tex]\( H ^{\prime \prime} \)[/tex] at its original position, and then shifting it directly down such that each point lowers by 2 units, resulting in the same shape but positioned slightly lower on the grid.
2. Dilation:
- [tex]\( H ^{\prime \prime} \)[/tex] is dilated by a scale factor of 2. This implies that each distance from the center of dilation (usually the origin) is doubled. Essentially, if you think about stretching the letter outward equally in all directions, the shape of [tex]\( H ^{\prime \prime} \)[/tex] is enlarged by twice its original size. For instance, each line segment in [tex]\( H ^{\prime \prime} \)[/tex] gets twice as long, and the spaces between segments also increase similarly.
3. Rotation:
- After dilation, the next transformation is a rotation of [tex]\( 180^\circ \)[/tex] clockwise. When the letter is rotated by 180 degrees clockwise, it turns around to face the opposite direction. Essentially, the top becomes the bottom, and the left side swaps with the right side.
4. Final Rotation:
- Finally, the letter is rotated [tex]\(90^\circ\)[/tex] counterclockwise. Visualize the letter at its current orientation post-180-degree rotation, and then simply rotate it by 90 degrees in the opposite direction of the clockwise movement, i.e., counterclockwise. This further changes the letter’s direction and orientation, aligning it with the final shape and positioning.
By carefully applying these transformations in sequence, the letter [tex]\( H \)[/tex] indeed matches its final form described as 'H' with respect to its shape and orientation on the grid.
