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Formula Used
A y-axis reflection reverses every horizontal coordinate. It preserves each vertical coordinate.
For a function, replace the independent variable with its opposite.
For an implicit relation, replace every occurrence of x.
How to Use This Calculator
- Select point, function, or polygon mode.
- Enter coordinates or a supported mathematical expression.
- Choose decimals, fractions, precision, and graph settings.
- Press the reflection button to process the input.
- Review tables, steps, graphs, symmetry, and measurements.
- Copy, print, or export the finished calculation.
Example Data Table
| Example | Original input | Reflected result | Key observation |
|---|---|---|---|
| Point A | (3, 5) | (−3, 5) | x changes sign. |
| Point B | (−2, 4) | (2, 4) | Negative x becomes positive. |
| Point C | (0, −1) | (0, −1) | A y-axis point stays fixed. |
| Linear function | y = 2x + 1 | y = −2x + 1 | The slope changes sign. |
| Quadratic function | y = x² + 3x | y = x² − 3x | Even powers remain unchanged. |
| Even function | y = x² | y = x² | The graph already has y-axis symmetry. |
Understanding Reflection Across the Y-Axis
What the transformation means
Reflection across the y-axis creates a horizontal mirror image. Every point moves to the opposite side. Its distance from the y-axis stays equal. The height of the point does not change. This rule makes the transformation predictable and easy to verify.
How coordinates change
An ordered pair contains horizontal and vertical information. The first value gives the horizontal position. The second value gives the vertical position. A y-axis reflection changes only the first value. Therefore, positive x-values become negative. Negative x-values become positive. Zero remains zero because it has no opposite side.
How functions change
A function graph consists of many coordinate points. Replacing x with negative x reflects all those points together. The new rule is written as f of negative x. Even powers often remain unchanged after simplification. Odd powers usually change their signs. Trigonometric and rational functions follow the same substitution rule.
Recognizing y-axis symmetry
A function has y-axis symmetry when reflection produces the same graph. Algebraically, this happens when f of negative x equals f of x. Such a function is called even. The graph of x squared is a familiar example. Constant functions are also even. A numerical test can suggest symmetry, but symbolic proof remains stronger.
Reflecting polygons and shapes
A polygon is reflected by transforming every vertex. The vertices stay connected in their original order. Side lengths remain equal because reflection is rigid. Angles, perimeter, and area also remain unchanged. However, clockwise orientation becomes counterclockwise. The centroid receives the same coordinate reflection rule.
Common mistakes to avoid
Do not change the y-coordinate during this reflection. That rule belongs to x-axis reflection. Do not negate the entire ordered pair. Doing so creates a rotation about the origin. Use parentheses when substituting negative x into powers. This prevents sign errors and preserves correct operation order.
Checking a completed result
Compare distances from the y-axis first. Original and reflected points should have equal absolute x-values. Their y-values should match exactly. A graph should look like a horizontal mirror image. For polygons, lengths and area should remain equal. These checks quickly reveal most entry mistakes.
Why graphing is useful
A graph turns the algebraic rule into a visible movement. Students can compare corresponding points directly. Teachers can demonstrate fixed points on the y-axis. Designers can inspect mirrored coordinates for technical layouts. The interactive controls also help explore large, small, or fractional values.
Reflection Compared With Other Transformations
| Transformation | Coordinate rule | What changes |
|---|---|---|
| Reflect over y-axis | (x, y) → (−x, y) | Horizontal coordinate |
| Reflect over x-axis | (x, y) → (x, −y) | Vertical coordinate |
| Reflect over origin | (x, y) → (−x, −y) | Both coordinates |
| Reflect over y = x | (x, y) → (y, x) | Coordinate positions swap |
| Rotate 180 degrees | (x, y) → (−x, −y) | Both signs change |