Question
\( \mathrm{J}(-7,7) \) Write an algebraic representation for the reflec
Ask by Lyons Wade. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To reflect the point \( J(-7, 7) \) across an axis or line, use the following rules:
1. **Across the x-axis:** \( J' = (-7, -7) \)
2. **Across the y-axis:** \( J' = (7, 7) \)
3. **Across the line \( y = x \):** \( J' = (7, -7) \)
4. **Across the origin:** \( J' = (7, -7) \)
Choose the rule that matches the reflection you want.
Solution
To create an algebraic representation for the reflection of the point \( J(-7, 7) \), we first need to specify **across which axis or line** the reflection is taking place. Common reflections include:
1. **Reflection Across the x-axis:**
- **Rule:** \( (x, y) \rightarrow (x, -y) \)
- **Application to \( J(-7, 7) \):**
\[
J' = (-7, -7)
\]
- **Explanation:** Only the y-coordinate changes its sign, flipping the point over the x-axis.
2. **Reflection Across the y-axis:**
- **Rule:** \( (x, y) \rightarrow (-x, y) \)
- **Application to \( J(-7, 7) \):**
\[
J' = (7, 7)
\]
- **Explanation:** Only the x-coordinate changes its sign, flipping the point over the y-axis.
3. **Reflection Across the Line \( y = x \):**
- **Rule:** \( (x, y) \rightarrow (y, x) \)
- **Application to \( J(-7, 7) \):**
\[
J' = (7, -7)
\]
- **Explanation:** The x and y coordinates are swapped, reflecting the point over the line \( y = x \).
4. **Reflection Across the Origin (Point Reflection):**
- **Rule:** \( (x, y) \rightarrow (-x, -y) \)
- **Application to \( J(-7, 7) \):**
\[
J' = (7, -7)
\]
- **Explanation:** Both coordinates change their signs, effectively rotating the point 180 degrees around the origin.
**Example: Reflection Across the x-axis**
Given the point \( J(-7, 7) \), reflecting it across the x-axis:
\[
J' = (-7, -7)
\]
**Algebraic Representation:**
\[
(x, y) \rightarrow (x, -y)
\]
This represents the general rule for reflecting any point across the x-axis.
---
**Choose the appropriate reflection rule based on the desired axis or line of reflection.**
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To reflect the point \( J(-7, 7) \) over the x-axis, you simply change the sign of the y-coordinate. The new point would then be \( J'(-7, -7) \). If you want to reflect over the y-axis, you change the sign of the x-coordinate, resulting in the point \( J''(7, 7) \). Both reflections will give you coordinates that are symmetrically opposite to the original point in their respective axes.
