Question: User: Reflections
Name: Cora Shtivelman
S1 S2 S3 S4
MEDIUM
Graph the image of the figure using the given transformations.
1) reflection across the y-axis
2) reflection across the x-axis
3) reflection across the x-axis
4) reflection across the y-axis
5) reflection across the y-axis
6) reflection across the y-axis
Graph shapes and position hints:
1) A quadrilateral/line figure on a coordinate grid with vertices marked and reflected across the y-axis; position_hint: top-left
2) A line segment/triangle-like figure on a coordinate grid reflected across the x-axis; position_hint: top-right
3) A multi-segment figure on a coordinate grid reflected across the x-axis; position_hint: center-left
4) A triangle on a coordinate grid reflected across the y-axis; position_hint: center-right
5) A triangle-like figure on a coordinate grid reflected across the y-axis; position_hint: bottom-left
6) A polygonal figure on a coordinate grid reflected across the y-axis; position_hint: bottom-right
1. **Problem Statement:**
We are asked to graph the images of various figures after reflecting them across either the $y$-axis or the $x$-axis.
2. **Reflection Rules:**
- Reflection across the $y$-axis: The $x$-coordinate of each point changes sign, i.e., $(x,y) \to (-x,y)$.
- Reflection across the $x$-axis: The $y$-coordinate of each point changes sign, i.e., $(x,y) \to (x,-y)$.
3. **Step-by-step for the first figure (top-left):**
- Given vertices of the quadrilateral/line figure: $(x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4)$.
- Reflect across the $y$-axis: new vertices are $(-x_1,y_1), (-x_2,y_2), (-x_3,y_3), (-x_4,y_4)$.
4. **Step-by-step for the second figure (top-right):**
- Given vertices of the line segment/triangle-like figure: $(x_1,y_1), (x_2,y_2), (x_3,y_3)$.
- Reflect across the $x$-axis: new vertices are $(x_1,-y_1), (x_2,-y_2), (x_3,-y_3)$.
5. **Step-by-step for the third figure (center-left):**
- Given vertices of the multi-segment figure: $(x_i,y_i)$ for $i=1,...,n$.
- Reflect across the $x$-axis: new vertices are $(x_i,-y_i)$ for all $i$.
6. **Step-by-step for the fourth figure (center-right):**
- Given vertices of the triangle: $(x_1,y_1), (x_2,y_2), (x_3,y_3)$.
- Reflect across the $y$-axis: new vertices are $(-x_1,y_1), (-x_2,y_2), (-x_3,y_3)$.
7. **Step-by-step for the fifth figure (bottom-left):**
- Given vertices of the triangle-like figure: $(x_1,y_1), (x_2,y_2), (x_3,y_3)$.
- Reflect across the $y$-axis: new vertices are $(-x_1,y_1), (-x_2,y_2), (-x_3,y_3)$.
8. **Step-by-step for the sixth figure (bottom-right):**
- Given vertices of the polygonal figure: $(x_i,y_i)$ for $i=1,...,m$.
- Reflect across the $y$-axis: new vertices are $(-x_i,y_i)$ for all $i$.
**Summary:**
- Reflection across the $y$-axis changes the sign of the $x$-coordinate.
- Reflection across the $x$-axis changes the sign of the $y$-coordinate.
Since the problem is about graphing transformations, the key is to apply these coordinate changes to each vertex of the figures.
**Final note:**
- The problem involves 6 distinct figures and transformations.
- Each reflection is applied as described above.