Reflection Rule Calculator Calculus

Reflection Rule:

\[ f(-x) \text{ represents reflection across the y-axis} \]

Reflected Function:

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1. What Is Reflection Rule in Calculus?

Reflection rules in calculus describe how functions transform when reflected across axes. The most common reflection is f(-x) which represents reflection across the y-axis, flipping the function horizontally.

2. How Reflection Transformation Works

Reflection transformations follow specific mathematical rules:

\[ \text{Y-axis reflection: } f(x) \rightarrow f(-x) \] \[ \text{X-axis reflection: } f(x) \rightarrow -f(x) \] \[ \text{Origin reflection: } f(x) \rightarrow -f(-x) \]

Where:

\( f(x) \) — Original function
\( f(-x) \) — Function reflected across y-axis
\( -f(x) \) — Function reflected across x-axis
\( -f(-x) \) — Function reflected through origin

Explanation: These transformations preserve the shape of the function while changing its orientation relative to the coordinate axes.

3. Types of Reflections

Details: Reflection across y-axis (horizontal flip) changes the x-coordinates to their opposites while keeping y-values the same. Reflection across x-axis (vertical flip) changes y-coordinates to their opposites while keeping x-values the same.

4. Using the Reflection Calculator

Tips: Enter your function using standard mathematical notation (e.g., x^2, sin(x), 2x+3). Select the type of reflection you want to apply. The calculator will show the transformed function.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between f(-x) and -f(x)?
A: f(-x) reflects the function across the y-axis (horizontal flip), while -f(x) reflects across the x-axis (vertical flip).

Q2: Can I reflect trigonometric functions?
A: Yes, the reflection rules apply to all types of functions including trigonometric, exponential, and polynomial functions.

Q3: How does reflection affect function properties?
A: Reflection may change symmetry properties. For example, an even function f(x) = f(-x) remains unchanged under y-axis reflection.

Q4: What is origin reflection?
A: Origin reflection is equivalent to reflecting across both axes simultaneously, which rotates the function 180 degrees around the origin.

Q5: Can I reflect piecewise functions?
A: Yes, but you need to apply the reflection transformation to each piece separately, considering how it affects the domain of each piece.