The different types of reflection functions primarily involve reflecting the graph of a function across either the x-axis or the y-axis.
Understanding Function Reflections
Function reflections are transformations that create a mirror image of the original function's graph. These transformations alter the position and orientation of the graph but not its shape or size. Based on the axis of reflection, there are two main types: vertical and horizontal reflections.
Vertical Reflection
A vertical reflection of a function produces a new graph that is a mirror image of the base or original graph about the x-axis.
- How it looks: Every point $(x, y)$ on the original graph of $y = f(x)$ is transformed to $(x, -y)$.
- Function Notation: If the original function is $f(x)$, the vertically reflected function is represented by $-f(x)$.
- Effect: This transformation flips the graph upside down. Positive y-values become negative, and negative y-values become positive, while the x-values remain unchanged.
Horizontal Reflection
A horizontal reflection of a function produces a new graph that is a mirror image of the base or original graph about the y-axis.
- How it looks: Every point $(x, y)$ on the original graph of $y = f(x)$ is transformed to $(-x, y)$.
- Function Notation: If the original function is $f(x)$, the horizontally reflected function is represented by $f(-x)$.
- Effect: This transformation flips the graph left-to-right. X-values on the right side of the y-axis move to the left, and x-values on the left side move to the right, while the y-values remain unchanged.
Summary Table
| Type of Reflection | Axis of Reflection | Transformation Rule | Function Notation | Effect on Graph |
|---|---|---|---|---|
| Vertical | x-axis | $(x, y) \to (x, -y)$ | $-f(x)$ | Flips upside down |
| Horizontal | y-axis | $(x, y) \to (-x, y)$ | $f(-x)$ | Flips left-to-right |
Understanding these two fundamental types of reflections is key to analyzing function transformations and their graphical representations.
