Reflecting a function in an equation depends on the axis of reflection. Here's how to write the equation for reflections across the x-axis and y-axis:
Reflection Across the x-axis
To reflect a function y = f(x) across the x-axis, you replace y with -y. Therefore, the new equation becomes:
- -y = f(x)
which can be rewritten as:
- y = -f(x)
Example:
If the original equation is y = x2, the reflection across the x-axis is y = -x2.
Reflection Across the y-axis
To reflect a function y = f(x) across the y-axis, you replace x with -x. Therefore, the new equation becomes:
- y = f(-x)
Example:
If the original equation is y = x3, the reflection across the y-axis is y = (-x)3 = -x3. Note that if the original equation was y = x2, the reflection across the y-axis is y = (-x)2 = x2, meaning the function is unchanged (symmetric).
Summary Table
| Reflection Axis | Transformation | Equation Change | Example Original: y = x+2 | Example Reflected |
|---|---|---|---|---|
| x-axis | (x, y) → (x, -y) | y becomes -y | y = x + 2 | y = -x - 2 |
| y-axis | (x, y) → (-x, y) | x becomes -x | y = x + 2 | y = -x + 2 |
In summary, reflecting a function involves changing either the y or x variable to its negative counterpart, depending on whether you are reflecting across the x-axis or y-axis, respectively.
