Reflections of shapes over the x-axis and y-axis on coordinate grids.
Four coordinate grid diagrams showing geometric shapes reflected over the lines y = 0 and x = 0, with red lines indicating the reflected images.
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Show Answer Key & Explanations
Step-by-step solution for: Reflection of 5 Vertices Over the x or y Axis (A)
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Show Answer Key & Explanations
Step-by-step solution for: Reflection of 5 Vertices Over the x or y Axis (A)
To solve these reflection problems, we need to understand how coordinates change when a shape is flipped (reflected) across an axis.
1. Reflection over the y-axis ($y = 0$)
* Rule: When you reflect a point over the vertical y-axis, the x-coordinate changes its sign (positive becomes negative, negative becomes positive), but the y-coordinate stays the same.
* Formula: $(x, y) \rightarrow (-x, y)$
* Visual Check: The red shape should look like a mirror image of the black shape across the vertical line in the middle. Points on the right side move to the left side at the same distance from the center, and vice versa. The height (up and down position) does not change.
2. Reflection over the x-axis ($x = 0$)
* Rule: When you reflect a point over the horizontal x-axis, the y-coordinate changes its sign, but the x-coordinate stays the same.
* Formula: $(x, y) \rightarrow (x, -y)$
* Visual Check: The red shape should look like a mirror image of the black shape across the horizontal line in the middle. Points above the line move below it, and points below move above. The horizontal position (left and right) does not change.
Verification of the Graphs:
* Top Left (Reflect over $y=0$):
* Take a vertex of the black hexagon, for example, the top-right corner at approximately $(3, 4)$.
* Apply rule $(-x, y)$: It becomes $(-3, 4)$.
* Looking at the red graph, the corresponding top-left corner is indeed at $(-3, 4)$. The reflection is correct.
* Top Right (Reflect over $y=0$):
* Take the right-most tip of the black quadrilateral at approximately $(4, 1)$.
* Apply rule $(-x, y)$: It becomes $(-4, 1)$.
* Looking at the red graph, the left-most tip is at $(-4, 1)$. The reflection is correct.
* Bottom Left (Reflect over $x=0$):
* Take the top-right corner of the black trapezoid at approximately $(3, 2)$.
* Apply rule $(x, -y)$: It becomes $(3, -2)$.
* Looking at the red graph, the bottom-right corner is at $(3, -2)$. The reflection is correct.
* Bottom Right (Reflect over $x=0$):
* Take the top-right corner of the black parallelogram at approximately $(4, 4)$.
* Apply rule $(x, -y)$: It becomes $(4, -4)$.
* Looking at the red graph, the bottom-right corner is at $(4, -4)$. The reflection is correct.
All four graphs correctly show the reflected images according to the mathematical rules for reflections.
Final Answer:
The images provided in the document are the correct solutions.
1. Top Left: The polygon is correctly reflected across the vertical y-axis. Each point $(x,y)$ has been mapped to $(-x,y)$.
2. Top Right: The quadrilateral is correctly reflected across the vertical y-axis. Each point $(x,y)$ has been mapped to $(-x,y)$.
3. Bottom Left: The trapezoid is correctly reflected across the horizontal x-axis. Each point $(x,y)$ has been mapped to $(x,-y)$.
4. Bottom Right: The parallelogram is correctly reflected across the horizontal x-axis. Each point $(x,y)$ has been mapped to $(x,-y)$.
1. Reflection over the y-axis ($y = 0$)
* Rule: When you reflect a point over the vertical y-axis, the x-coordinate changes its sign (positive becomes negative, negative becomes positive), but the y-coordinate stays the same.
* Formula: $(x, y) \rightarrow (-x, y)$
* Visual Check: The red shape should look like a mirror image of the black shape across the vertical line in the middle. Points on the right side move to the left side at the same distance from the center, and vice versa. The height (up and down position) does not change.
2. Reflection over the x-axis ($x = 0$)
* Rule: When you reflect a point over the horizontal x-axis, the y-coordinate changes its sign, but the x-coordinate stays the same.
* Formula: $(x, y) \rightarrow (x, -y)$
* Visual Check: The red shape should look like a mirror image of the black shape across the horizontal line in the middle. Points above the line move below it, and points below move above. The horizontal position (left and right) does not change.
Verification of the Graphs:
* Top Left (Reflect over $y=0$):
* Take a vertex of the black hexagon, for example, the top-right corner at approximately $(3, 4)$.
* Apply rule $(-x, y)$: It becomes $(-3, 4)$.
* Looking at the red graph, the corresponding top-left corner is indeed at $(-3, 4)$. The reflection is correct.
* Top Right (Reflect over $y=0$):
* Take the right-most tip of the black quadrilateral at approximately $(4, 1)$.
* Apply rule $(-x, y)$: It becomes $(-4, 1)$.
* Looking at the red graph, the left-most tip is at $(-4, 1)$. The reflection is correct.
* Bottom Left (Reflect over $x=0$):
* Take the top-right corner of the black trapezoid at approximately $(3, 2)$.
* Apply rule $(x, -y)$: It becomes $(3, -2)$.
* Looking at the red graph, the bottom-right corner is at $(3, -2)$. The reflection is correct.
* Bottom Right (Reflect over $x=0$):
* Take the top-right corner of the black parallelogram at approximately $(4, 4)$.
* Apply rule $(x, -y)$: It becomes $(4, -4)$.
* Looking at the red graph, the bottom-right corner is at $(4, -4)$. The reflection is correct.
All four graphs correctly show the reflected images according to the mathematical rules for reflections.
Final Answer:
The images provided in the document are the correct solutions.
1. Top Left: The polygon is correctly reflected across the vertical y-axis. Each point $(x,y)$ has been mapped to $(-x,y)$.
2. Top Right: The quadrilateral is correctly reflected across the vertical y-axis. Each point $(x,y)$ has been mapped to $(-x,y)$.
3. Bottom Left: The trapezoid is correctly reflected across the horizontal x-axis. Each point $(x,y)$ has been mapped to $(x,-y)$.
4. Bottom Right: The parallelogram is correctly reflected across the horizontal x-axis. Each point $(x,y)$ has been mapped to $(x,-y)$.
Parent Tip: Review the logic above to help your child master the concept of reflection of shapes worksheet.
