Reflection

TL;DR

Reflection is like looking in a mirror, where a shape flips over a line called the line of reflection, creating a mirrored image. Each point in the original shape is the same distance from the line of reflection as its corresponding point in the reflected image. The reflected image is the same size and shape as the original but is oriented differently.

1. The Mental Model

Imagine folding a piece of paper along a line and pressing an ink drawing from one side onto the other. The new image you see is a reflection of the original. The fold line is your line of reflection, and everything on one side gets perfectly mirrored on the other.

2. The Core Material

When you reflect a shape, every point on that shape has a corresponding point on the reflected image. The line of reflection acts as a perpendicular bisector for the segment connecting any point to its reflected counterpart.

Reflecting over the x-axis

Photo by Jerry Apples on Pexels

If a point has coordinates (x, y), reflecting it over the x-axis changes its y-coordinate to its opposite, but keeps the x-coordinate the same.
The rule is: (x, y) --> (x, -y).

Reflecting over the y-axis

Photo by Jerry Apples on Pexels

If a point has coordinates (x, y), reflecting it over the y-axis changes its x-coordinate to its opposite, but keeps the y-coordinate the same.
The rule is: (x, y) --> (-x, y).

Reflecting over the line y = x

Photo by Onur Yumlu on Pexels

If a point has coordinates (x, y), reflecting it over the line y = x swaps its x and y coordinates.
The rule is: (x, y) --> (y, x).

Reflecting over the line y = -x

Photo by Nothing Ahead on Pexels

If a point has coordinates (x, y), reflecting it over the line y = -x swaps its x and y coordinates and changes both of their signs.
The rule is: (x, y) --> (-y, -x).

Reflecting over a horizontal line (y = k) or a vertical line (x = k)

These reflections are a bit more general.
* For a horizontal line y = k: The x-coordinate stays the same. The new y-coordinate is 2k - y.
Rule: (x, y) --> (x, 2k - y).
* For a vertical line x = k: The y-coordinate stays the same. The new x-coordinate is 2k - x.
Rule: (x, y) --> (2k - x, y).

Here's how to think about the different types of reflections:

graph TD
    A["Starting Point (x, y)"] --> B{{"Reflect over what?"}}
    B --> C["x-axis"]
    C --> D["(x, -y)"]
    B --> E["y-axis"]
    E --> F["(-x, y)"]
    B --> G["Line y = x"]
    G --> H["(y, x)"]
    B --> I["Line y = -x"]
    I --> J["(-y, -x)"]
    B --> K["Horizontal line y = k"]
    K --> L["(x, 2k - y)"]
    B --> M["Vertical line x = k"]
    M --> N["(2k - x, y)"]

3. Worked Example

Let's reflect a triangle with vertices A(1, 2), B(3, 5), and C(4, 1) over the line y = -1.

1. Identify the type of reflection: This is a reflection over a horizontal line y = k, where k = -1.
2. Apply the rule: The rule for reflecting over y = k is (x, y) --> (x, 2k - y).
3. Calculate the new coordinates for each vertex:
   • For A(1, 2):
     x-coordinate remains 1.
     New y-coordinate = 2 * (-1) - 2 = -2 - 2 = -4.
     So, A' is (1, -4).
   • For B(3, 5):
     x-coordinate remains 3.
     New y-coordinate = 2 * (-1) - 5 = -2 - 5 = -7.
     So, B' is (3, -7).
   • For C(4, 1):
     x-coordinate remains 4.
     New y-coordinate = 2 * (-1) - 1 = -2 - 1 = -3.
     So, C' is (4, -3).

The reflected triangle A'B'C' has vertices at A'(1, -4), B'(3, -7), and C'(4, -3).

4. Key Takeaways

• A reflection flips a shape across a line of reflection.
• The reflected image is congruent (same size and shape) to the original.
• The orientation of the image is reversed compared to the original.
• Each point and its reflected image are equidistant from the line of reflection.
• Common reflection lines are the x-axis, y-axis, y=x, and y=-x.
• For lines like y=k or x=k, use the specific formulas to find new coordinates.

Common mistakes to avoid:
- Confusing reflection over y=x with y=-x.
- Forgetting to change the sign of the coordinate that's being reflected.
- Incorrectly calculating 2k - y or 2k - x for general line reflections.
- Not reflecting all vertices of a shape.

5. Now Try It

Reflect a rectangle with vertices P(-2, 3), Q(4, 3), R(4, -1), and S(-2, -1) over the y-axis.

What to do: List the original coordinates, apply the correct reflection rule to each vertex, and then list the coordinates of the reflected rectangle P'Q'R'S'.

What success looks like: You'll have four new coordinate pairs, each with the x-coordinate's sign flipped and the y-coordinate unchanged, matching the rule (x, y) --> (-x, y).