Sure! Let's go through the process step by step to create a function that represents the point [tex]\((x, y)\)[/tex] being reflected across the [tex]\(y\)[/tex]-axis.
### Reflection Across the [tex]\(y\)[/tex]-Axis
When you reflect a point across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate of the point changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
For a point [tex]\((x, y)\)[/tex]:
- The reflected point will be [tex]\((-x, y)\)[/tex].
In mathematical terms, this transformation can be described by the function:
[tex]\[ F(x, y) = (-x, y) \][/tex]
### Step-by-Step Solution
1. Define the Input Coordinates:
- Let the original coordinates of the point be [tex]\((x, y)\)[/tex].
2. Reflect Across the [tex]\(y\)[/tex]-Axis:
- To find the reflected point, negate the [tex]\(x\)[/tex]-coordinate: [tex]\(-x\)[/tex].
- The [tex]\(y\)[/tex]-coordinate remains the same: [tex]\(y\)[/tex].
3. Formulate the Reflected Coordinates:
- The new coordinates after reflection will be [tex]\((-x, y)\)[/tex].
### Example
Let's work through an example to illustrate this:
Original Point:
- Suppose the original point is [tex]\((3, 4)\)[/tex].
Reflection Process:
- Negate the [tex]\(x\)[/tex]-coordinate: [tex]\(-3\)[/tex].
- Keep the [tex]\(y\)[/tex]-coordinate unchanged: [tex]\(4\)[/tex].
Reflected Point:
- The reflected point will be [tex]\((-3, 4)\)[/tex].
### General Function
To generalize this process for any point [tex]\((x, y)\)[/tex], we define a function [tex]\(F\)[/tex] as follows:
[tex]\[ F(x, y) = (-x, y) \][/tex]
So, if you are given any point [tex]\((x, y)\)[/tex] and you want to find the coordinates of the point after it has been reflected across the [tex]\(y\)[/tex]-axis, you simply apply the transformation provided by the function [tex]\(F\)[/tex].
### Conclusion
The function [tex]\(F(x, y) = (-x, y)\)[/tex] represents the point [tex]\((x, y)\)[/tex] being reflected across the [tex]\(y\)[/tex]-axis. This transformation changes the sign of the [tex]\(x\)[/tex]-coordinate and keeps the [tex]\(y\)[/tex]-coordinate unchanged.
