Flipping a square root function horizontally is achieved by multiplying the variable inside the square root by -1.
Understanding function transformations helps us manipulate graphs of functions like the square root. Transformations can include shifting, stretching, compressing, and reflecting the graph.
Understanding Horizontal Reflection
A horizontal reflection, or flip across the y-axis, changes the sign of the input variable (x).
How to Apply the Transformation
To flip the basic square root function, (f(x) = \sqrt{x}), horizontally:
- Identify the independent variable: This is (x) in the standard function (f(x) = \sqrt{x}).
- Replace the variable with its negative: Substitute (x) with (-x).
The resulting function represents the horizontally flipped graph.
Transformed Function Formula
The horizontally flipped version of (f(x) = \sqrt{x}) is:
(g(x) = f(-x) = \sqrt{-x})
Example Transformation
Let's compare the domain and a few points for (f(x) = \sqrt{x}) and (g(x) = \sqrt{-x}):
| Function | Domain | Example Points |
|---|---|---|
| (f(x) = \sqrt{x}) | (x \ge 0) | (0, 0), (1, 1), (4, 2) |
| (g(x) = \sqrt{-x}) | (x \le 0) | (0, 0), (-1, 1), (-4, 2) |
Notice that for (g(x) = \sqrt{-x}), the input (x) must be less than or equal to 0 for (-x) to be non-negative, which is required for the square root of a real number. This restricts the domain and reflects the graph across the y-axis.
Context of Radical Function Transformations
Transforming radical functions is a common practice in algebra. While flipping horizontally involves manipulating the variable inside the radical, other transformations, such as vertical shifts, are applied outside the radical. As noted in the reference, adding or subtracting a value outside the square root function (e.g., (y = \sqrt{x} + c)) results in a vertical shift of the graph.
In summary:
- To flip a square root function horizontally, change (x) to (-x) inside the radical.
- This transforms (y = \sqrt{x}) into (y = \sqrt{-x}).
- The domain of the function changes from (x \ge 0) to (x \le 0).
