In mathematics, a translation is a transformation that slides a figure or point from one location to another without changing its size, shape, or orientation. When plotting points using the rule $(x+a, y+b)$, 'a' represents the horizontal shift (left or right), and 'b' represents the vertical shift (up or down). For example, if we have the rule $(x+2, y-3)$, it means we shift each point 2 units to the right and 3 units down.
Understanding translations is crucial in geometry and helps visualize how objects move in space. By applying these rules, we can easily determine the new coordinates of any point after translation. This concept is fundamental for more advanced topics such as vector transformations and coordinate geometry.
Match the terms with their correct definitions:
| Terms | Definitions |
|---|---|
| 1. Translation | A. A transformation that changes the size of a figure. |
| 2. Coordinate | B. A transformation that flips a figure over a line. |
| 3. Transformation | C. A location specified by an ordered pair $(x, y)$. |
| 4. Dilation | D. A transformation that slides a figure without changing its size or shape. |
| 5. Reflection | E. A general term for altering the position or form of a figure. |
Complete the following paragraph with the correct words:
A __________ is a transformation that moves every point of a figure the same distance in the same __________. The rule $(x+a, y+b)$ represents a translation where 'a' indicates the __________ shift and 'b' indicates the __________ shift.
Explain, in your own words, how the values of 'a' and 'b' in the translation rule $(x+a, y+b)$ affect the position of a point on a coordinate plane. Provide an example to illustrate your explanation.
In mathematics, a translation shifts every point of a figure or a space by the same distance in a given direction. When we're working with coordinate points, a translation rule (x+a, y+b) means we're adding 'a' to the x-coordinate and 'b' to the y-coordinate of each point. This moves the point 'a' units horizontally and 'b' units vertically. If 'a' is positive, the point moves to the right; if negative, to the left. Similarly, if 'b' is positive, the point moves up; if negative, down. Understanding this concept allows us to predict and perform transformations on shapes and graphs easily.
For example, if we have a point (2, 3) and we apply the translation rule (x+1, y-2), the new point would be (2+1, 3-2) which simplifies to (3, 1). This shows a shift of 1 unit to the right and 2 units down.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Translation | A. The original point before a transformation. |
| 2. Pre-image | B. A transformation that slides a figure without changing its size or orientation. |
| 3. Image | C. The coordinate that represents vertical position on a graph. |
| 4. X-coordinate | D. The new point after a transformation. |
| 5. Y-coordinate | E. The coordinate that represents horizontal position on a graph. |
Answers: 1-B, 2-A, 3-D, 4-E, 5-C
A _________ shifts every point of a figure by the same distance in a given _________. The translation rule (x+a, y+b) means we add 'a' to the _________-coordinate and 'b' to the _________-coordinate. If 'a' is _________, the point moves to the right.
Answers: translation, direction, x, y, positive
Explain how the translation rules $(x+2, y-3)$ and $(x-3, y+2)$ affect a point differently and provide a real-world example where such translations might be useful.
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