Geometry Worksheets | Transformations Worksheets - Free Printable

Since I can't view or access images directly, I’ll help you solve the reflection problems based on your description of the image. However, since the image contains six separate reflection tasks (with shapes on coordinate grids), I'll explain how to solve each type of reflection and provide a general method for each.

Let’s go through each problem one by one and explain how to reflect the given figure across the specified line. If you can describe the coordinates of the original shape in each case, I can give exact answers. But here's how to approach each:

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1) Reflection Across Line y = -x

Rule: To reflect a point $(x, y)$ across the line $y = -x$, use the transformation:
$$
(x, y) \rightarrow (-y, -x)
$$

Steps:
- Take each vertex of the shape.
- Apply the rule above.
- Plot the new points and connect them.

Example:
If a point is at $(2, 3)$, its reflection is $(-3, -2)$.

> 🔍 *Look at the green shape in the first graph — it appears to be an "L" shape. Identify its vertices, apply the rule, and draw the reflected image.*

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2) Reflection Across Line y = x

Rule: Reflecting over $y = x$ swaps the coordinates:
$$
(x, y) \rightarrow (y, x)
$$

Steps:
- For each point, swap $x$ and $y$.
- Plot the new points.

Example:
Point $(4, 1)$ becomes $(1, 4)$.

> 🔍 The triangle in this graph likely has vertices like $(-2, 3), (-1, 1), (-3, 1)$ — reflect each using $(x, y) \to (y, x)$.

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3) Reflection Across the Line x = 1

This is a vertical line at $x = 1$. To reflect across it:

Rule:
The distance from a point to the line $x = 1$ is $|x - 1|$. The reflected point will be the same distance on the other side.

So:
$$
(x, y) \rightarrow (2 - x, y)
$$

Why? Because the midpoint between $x$ and $2 - x$ is $1$:
$$
\frac{x + (2 - x)}{2} = 1
$$

Example:
Point $(3, 2)$ → $ (2 - 3, 2) = (-1, 2) $

> 🔍 The green rectangle is likely centered near $x = 2$. Use the formula to reflect each corner.

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4) Reflection Across the y-axis

Rule:
$$
(x, y) \rightarrow (-x, y)
$$

Just change the sign of the $x$-coordinate.

Example:
$(3, 4)$ → $(-3, 4)$

> 🔍 The triangle is probably in the first quadrant. Reflect each vertex across the y-axis (flip left-right).

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5) Reflection Across the Line y = 2

This is a horizontal line at $y = 2$. To reflect across it:

Rule:
Distance from point to line is $|y - 2|$. Reflected point is same distance below/above.

So:
$$
(x, y) \rightarrow (x, 4 - y)
$$

Because midpoint: $\frac{y + (4 - y)}{2} = 2$

Example:
Point $(2, 5)$ → $(2, 4 - 5) = (2, -1)$

> 🔍 The parallelogram is above $y = 2$. Reflect each vertex down to mirror it.

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6) Reflection Across the x-axis

Rule:
$$
(x, y) \rightarrow (x, -y)
$$

Change the sign of the $y$-coordinate.

Example:
$(2, 3)$ → $(2, -3)$

> 🔍 The "staircase" shape is above the x-axis. Reflect each point downward.

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✔ General Steps to Solve Each Problem:

1. Identify the vertices of the original shape.
2. Apply the correct reflection rule based on the line.
3. Plot the reflected points.
4. Connect the points in the same order as the original.

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📌 Tips:

- Use graph paper or grid lines carefully.
- Double-check distances from the line of reflection.
- Label the reflected image clearly.

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If you can describe the coordinates of the original figures (e.g., list the vertices of each shape), I can give you the exact reflected coordinates for each problem.

Would you like to do that now? For example:
> “For problem 1, the L-shape has points at (-2,-1), (-2,-2), (-1,-2), (-1,-3), (0,-3), (0,-2)” — then I can compute the rest.