Reflection (Algebraic Transformations)
See It in Action
Two examples of the idea done right—and one common look-alike that is not right.
EXAMPLE
EXAMPLE
NON-EXAMPLE
Definition
In Algebra 1, a reflection refers to a transformation that "flips" a graph over a specific line, creating a mirror image of the original graph. Reflections change the orientation of the graph, but they do not alter its size or shape.
Key Types of Reflections:
1. Reflection over the x-axis:
When a function f(x) is reflected over the x-axis, it is transformed to -f(x).
This means that for every point on the graph, the y-coordinate changes sign, while the x-coordinate stays the same. The graph is flipped upside down.
Example:
If f(x) = x^2, the reflection of this over the x-axis is -f(x) = -x^2, which flips the graph of x^2 upside down.
2. Reflection over the y-axis:
When a function f(x) is reflected over the y-axis, it is transformed to f(-x).
This means that for every point on the graph, the x-coordinate changes sign, while the y-coordinate stays the same. The graph is flipped horizontally.
Example:
If f(x) = x^2, the reflection of this over the y-axis is f(-x) = (-x)^2 = x^2. For this specific example, the graph is the same because x^2 is symmetric with respect to the y-axis. But for other functions, such as f(x) = x, f(-x) would create a mirrored graph.
Why Does This Happen?
Reflection over the x-axis: The negative sign in front of f(x) flips all the y-coordinates, effectively turning the graph upside down.
Reflection over the y-axis: The negative inside the function f(-x) flips the graph horizontally by changing the x-coordinates. This causes the graph to mirror across the y-axis.
Summary:
A reflection in algebra is a transformation that flips a graph over a line (either the x-axis or y-axis) to create a mirror image of the original graph. Reflections change the orientation of the graph but maintain its size and shape.